Stats 2

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when talking about a confidence level say away from

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probability

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bad interpretations of confidence interval is

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percent of time and

probability

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good interpretation of confidence interval is

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confidence level

actual confidence level

saying what mew is

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margin of error

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mostly that x bar or your statistic could differ form mew for the middle confidence level

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good interpretation of confidence level

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is

confident in the procedure

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statistical inference

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drawing conclusions about a popultaion parameter using a sample statistic. There are two types

1. Estimating the value of a parameter with a confidence interval

2. Assessing evidence for a claim about a parameter value using a test of significance

using a statistic to talk about a parameter

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statistically significant

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an observed effect that is too large to plausibly be due to chance variation

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observed statistic

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claimed parameter value

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large observed effect

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has a small probability

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small observed effect

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has a big probability

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test of significance

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determining if an observed effect is statistically significant

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confidence level is about the

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procedure

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confidence level needs to say

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to say something like the percent of these intervals created in this way contain mew or the mean of the population

also confident in the procedure or the process

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the purpose of the confidence interval is to

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provide a range of reasonable values for mew

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we use the word probability

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before a sample is taken

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we use the word confidence

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after the sample is taken

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good interpretation of confidence interval needs to have

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a confidence level and that is always a percentage

and

needs to have an actual confidence interval which is like the range where you think mew looks like between two different numbers. a lower number and a higher number and you think mew is somewhere in there

and

saying what the mew is in words like the true mean or mean tied in bushes per acre

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bad interpretation of confidence interval

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bad things are a

percent of time

and

probability

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the main purpose of a confidence interval is

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is to catch mew in between two numbers

Front 20

good interpretations of the confidence level is

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we are confident in the procedure

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Ok

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And is

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a null hypothesis is

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H O, ho

the statement of no difference or no change

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test of significance is when

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someone says I think Mew is this, and then we are going to test it.

a procedure used to assess the evidence against a claim about the value of a parameter

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the pee value

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look for extreme or big, or as small or smaller, or as big or bigger,

the probability of obtaining a test statistic as extreme or more extreme than observed if HOe is true

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alternative hypothesis

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the symbol is Ha

and it is a statement of difference or change

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a test of significance is

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a procedure used to assess the evidence against a claim about the value of a parameter

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P value is

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probability of obtaining a test statistic as extreme or more extreme than observed if HOE is true

Front 28

If Pee is low then

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reject HOE

accept Ha

it is statistically significant

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if Pee is High then

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fail to reject the HOE

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are the results of an experiment statistically significant only if a random sample or random allocation of treatments occurred ?

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yes it is statistically significant

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before calculation a confidence interval, one must

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always check that the data are from an simple random sample and that the data are normal

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a significance level

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the symbol is A

gives us a point where we start to declare observed effects statistically significant.

Front 33

how to find the z start

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on the table

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if pee is high then

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fail to reject hoe

insufficient evidence, to accept Ha,

and not statistically significant

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what does a confidence level mean

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it is just the percentage, and when it talks about confidence we are confident in the procedure or the process

Front 36

what is the sixty eight, ninety five, ninety nine point seven rule?

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one standard deviation is sixty eight and so on

Front 37

statistically significant is

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when the pee value is lower than A

and

the observed effect is too large to be due to chance.

Front 38

if you want to look up the pee value for a less than, then,

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the table always gives you less than so less than is just normal, or what whatever shows up first on the table

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if you want to look up the pee value for a greater than then,

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you look up the opposite sign on the table

Front 40

to look up the pee value for the differs then you,

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always look at the negative, and then multiply it by two.

Front 41

when I'm thinking about the test of significance I'm thinking

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I am going to get a pee value and then compare it to alpha.

Front 42

definition of test of significance

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asses the evidence about the claim for a parameter

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to do the test of significane you need to

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first, stat the hypotheses

second, find the test statistic, or the z score

third, take the test of significance and get the pee value

fourth, make a prediction, conclusion, compare pee value to alpha, is it statistically significant? should I reject Hoe? should I accept HA?

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H A

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alternative hypothesis

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to do a test of significance we need

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random and normal

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the null hypotheses

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HOE

mew equals

it is always what your mew is

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if the pee is high then

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fail to reject the hoe

and there is insufficient evidence to accept HA

not statistically significant because your not making a change.

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its statistically significant if

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you make a change!!!!!!!

Front 49

if pee is high then that means it is higher than

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alpha

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if pee is low that means it is lower than

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alpha

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if pee is low then that means

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it is lower than alpha

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if pee is high then that means

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pee is higher than alpha

Front 53

the significance level symbol is

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alpha or the infinity looking sign

and

it gives us a point where we start to declare observed effects statistically significant

and, subjectively chosen by researcher before collecting data,

and, equals the risk of rejecting hoe when hoe is actually true

and is typically small, usually point zero five or smaller

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the null hypothesis is

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the symbol is HOE

and this is a statement of no difference or no change in the parameter value.

and it gives a parameter as equal to the claimed value with an equals sign.

an example is Hoe is mew equals mew with a zero or Hoe is mew equals twenty three

and the null hypothesis is the claim that you want to find evidence against

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what is the null hypothesis symbol

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it is HOE

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if Pee is low then

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reject the hoe! and accept HA

Front 57

do a tee test when

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do a tee test when you don't know z score or don't know sigma

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for a tee test

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it must be random, it must be normal, and you need sigma

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for a zee test

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needs random, needs normal, but the rules on normal have changed a little here, if it tells you it is normal than it is, but it also has to be bigger than forty, but if it is less than forty as long as there is no skewness or outliers then we say close enough, and it is normal

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robust is

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only applies to normal, a statistical procedure is robust if confidence level or pee value does not change very much when conditions are not met

tee is never robust with respect to randomization

population normally distributed,

tee is robust with respect to normality if, n is less than forty and there are no outliers or strong skewness

n is greater than forty since we can apply the central limit theorem

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for a matched pairs we need

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everyone has to have two treatments

identical twins

pre and post

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if the question says do a test of significance then it has to be

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a tee test

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matched pairs design is the most common test

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of a tee test

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a clue phrase for matched pairs is going to be

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the mean difference! this is a very important phrase

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with respect to matched pairs we want to see

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we want to see what the difference is between the two outcomes

add up all the difference and you get the mean difference

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the d bar

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is the same as x bar, and x bar is the sample mean so the d bar is the sample mean

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your degrees of freedom is

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n minus one

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a matched pairs tee test is

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analysis of data from a matched pairs design is the most common use of a one sample tee test

when performing a tee test on a matched pairs design we compare the differences

- we find the difference between each pair

the difference is the response with treatment one minus the response with treatment two

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if its matched pairs mew is always

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mew is always zero

Front 70

for a tee test we need

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we need random and normal

the best way to get normal is if it tells you, then you try the central limit theorem

if n is less than thirty, you would have to graph it and see if there is no skewness or outliers

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we only do matched pairs on the

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tee

Front 72

to get a pee value on a tee test you

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need to get your degrees of freedom

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to get you=r degrees of freedom you

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do n or the sample minus one

and remember its all about the pairs or the matches

Front 74

if pee is high then

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fail to reject hoe

because insufficient evidence to accept it

not statistically significant

and if pee is high then mew is in our confidence interval on two sided

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matched pairs

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eveeryone gets two treatemetns

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z test will tell you sigma but it won't tell you for a

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tee test

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if its matched pairs then mew is always

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zero

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matched pairs characteristics

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everyone gets two treatments

identical twins

pre and post

and look for the mean difference

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tee test characteristics are

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it needs to be random

it needs to be normal

and its normal if it says it is, or if greater than forty, or if n is less than for it is normal but it has to be without outliers and skewness

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the tee star can also be

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the critical values

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zee test characteristics are

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it must say random

it must be nor normal, and for this to be normal the qualifications are different the the tee test for the zee test to be normal it must either say it is normal or n must be greater than thirty

the zee test will also give you sigma

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for our confidence interval we are always trying to catch

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mew and mew is always zero

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the tee test is like

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fatter and flatter so its not standard normal

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All statistical procedures require data to be collected either randomly in a survey or with randomization in an experiment.

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this is true

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single peak is just

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one peak

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What is the most important condition for all statistical inferential procedures?

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The data must be appropriately collected using randomization.

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if its close enough or not really skewed or has no outliers then we can call that

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robust

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a completely randomized design is

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Individuals are randomly divided into two equally sized groups and each group is assigned to one of two treatments by the researcher.

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for experiments random is

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always good

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we are just interested in the

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difference for matched pairs

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s is the

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sample standard deviation

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whats a type one error

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reject HOE when you should

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a type two error is when you

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believe hoe when you shouldn't

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power is

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one minus beta,

making a correct change

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a z test is

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random, or says simple random sample,

it says its normal or n is greater than thirty

it also tells us sigma

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two sample T- Test is

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the difference of means

has to be random, some kind of random

and needs some kind of normal, it can tell you its normal, the population is normal, for tee test N has to be bigger than forty so that is n one plus n two has to be bigger than forty. Your total sample size has to be bigger than forty. if it is under forty and there is no skewness or outliers than you still can do it.

Also this test has to be independent

the groups have to be separate, no body can be in both groups. different medicines for each group

the null hypothesis, is no difference in population means or treatment means

it is plural, has two means

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matched pairs is

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everybody gets the same treatment

Front 98

for the two sample tee test

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is one mean is greater than the other or means are not equal.

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z is just

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one sample

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teeee is

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two samples

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any time you do a two sample test the

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the null hypothesis is they are the same

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if p is low then

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reject the hoe

and accept HA

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if peeee is high then

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not statistically significant

fail to reject the Hoe,

and insufficient evidence to accept Ha

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if pee is high then

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fail to reject the hoe

and insufficient evidence

not statistically significant

mew is in the confidence interval

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to compare the difference in means you

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minus

so mew one minus mew two

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In words, how do we describe x¯1−x¯2?

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The difference between two sample means

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the hypothesis cannot have anything to do with

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x bars so throw them out!

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independent is

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just for the teee test

when you have medicines assigned for two different types of groups

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how to tell if its significant or not?

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either peeee is low or peeeeee is high

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a peeee value is

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the probability of seeing a test statistic as extreme or more extreme

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if it is significant

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peee value is less than alpha

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if we have matched pairs data then

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use a matched pairs experiment

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One of the conditions that must be met for a two-sample t procedure is

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that the two samples are independent (separate).

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anova is

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the analysis of variance

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for anova we get an

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f test statistic

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for the nova there are

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there or more for the hoe hypothesis

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for the nova the ha, or alternative hypothesis there are

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at least one of the means is different or they might say not all means are the same

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there is never an x bar or a d bar in

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any of the hypothesis ever

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one sample tee conditions are

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two conditions

random

normal

for a tee test to get normal there are three ways

one, it tells you

two, n is bigger than forty

three, if n is less than forty than you need to check for outliers and skewness, need to check by graphing it

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for a two sample tee test the characteristics are

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three conditions

random,

normal,

and for the normal you add up both groups to get a total number

and it has to be independent, which means that the groups are separate,

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for a nova test

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four conditions

random,

normal, add up all the people in all the groups, it could be used for three or more groups

all your groups have to be independent.

and the new condition for nova is called equal variance, which is take the biggest s and divide it by the smallest s and that has to be less than two, and if its not less than two, then it is not met.

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s is the

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sample standard deviation

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zee test

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random

normal, for this one, it can tell you it is normal, or it has to be thirty or above, that is the only ways it can be normal

also it will tell sigma, if it doesn't tell you sigma then it can't be a zee test

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the significance level is

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alpha

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confidence level is like

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a percentage like ninety percent

Front 126

nova or analysis of variance is only used for

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you can only take the mean of quantitative data so nova is only used to compare means from quantitative response variables. A chi squared test will be used later to compare proportions of categorical data sets

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the nova mean null hypothesis mean is

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the null hypotheses for nova is that the means are always the same

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HOE is the

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null hypothesis

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HA is the

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alternative hypothesis

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for two sided tests if pee is low then

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it is outside the interval

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three or more groups is

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nova

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f equals is the

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test statistic

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if pee is low then mew is not within the

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confidence interval

Front 134

What happens when the equality of three means is tested by performing three separate two-sample t tests?

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The overall α for all tests combined is inflated.

alpha would inflate

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True or false: We can use Analysis of Variance or ANOVA to compare three or more population means.

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true

Front 136

One of the conditions for ANOVA is equal variances. How do we check for equal variances?

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Make sure that the largest standard deviation divided by the smallest standard deviation is less than two.

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What type of response variable is required for an analysis of variance (ANOVA) procedure--categorical or quantitative?

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quantitative

Front 138

Which procedure should be used to compare three or more means?

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Analysis of variance (ANOVA) for comparing three or more means when the response variable is quantitative.

Front 139

to check nova we need

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at least three groups and see if the response variables are quantitate

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categorical is like a

Back 140

yes or a no question

Front 141

Many studies suggest that there is a link between exercise and healthy bones and that exercise stresses bones, causing them to strengthen. One completely randomized experiment examined the effect of jumping on the bone density of growing rats with three treatments—10 rats in a control with no jumping, 10 rats in a low-jump condition and 10 rats in a high-jump condition. After 8 weeks of 10 jumps per day, 5 days per week, the bone density of the rats (expressed in mg/cm3) was measured for each rat. Can we use ANOVA (analysis of variance) to analyze these data?

Back 141

Yes, because the response variable is quantitative and there are three treatment groups.

Front 142

Researchers explored the question, “Do poets die young?” They randomly sampled deceased women writers in North America in three categories: novels, poems, and nonfiction. For each woman they found their age at time of death. Should the ANOVA procedure be used to answer their research question?

Back 142

Yes, because there are three groups for the explanatory variable and the response variable is quantitative.

Front 143

The results of the study described in question 7 were statistically significant with the mean age of death for those who wrote poetry significantly lower than the mean age at death for the other two groups. Can the results of that study be used to conclude that writing poetry causes early death?

Back 143

No, because the study was observational.

Front 144

A recent study of 865 college students found that 42.5% had student loans. In summarizing the results of the study, students were classified as to their field of study (explanatory variable) and whether they have a student loan (response variable). Can we use ANOVA (analysis of variance) to anallyze these data? (To answer this question, you have to decide whether the response variable is categorical or quantitative.)

Back 144

No, because although there are three or more groups of students, the response variable is categorical.

Front 145

What is the purpose of a confidence interval?

Back 145

To provide a range of reasonable values for the unknown population parameter

Front 146

What is the name for the quantity z*(σ / √n)?

Back 146

Margin of error

z*(σ / √n) is the margin of error for estimating μ with σ known. (σ / √n) is the standard deviation of x̄ and the confidence interval is ( x̄ – z*(σ / √n), x̄ + z*(σ / √n)).

Front 147

Suppose that you give the NAEP (National Assessment of Educational Progress) test to an SRS of 1000 people from a large population in which the scores have mean μ = 280 and standard deviation σ = 60. The mean x̄ of the 1000 scores will vary if you take repeated samples.The sampling distribution of x̄ is approximately Normal. It has mean μ = 280 and standard deviation (σ / √n). According to the 68-95-99.7 rule, about 95% of all the values of x̄ fall within the margin of error of the mean μ. What is the value of the margin of error?

Back 147

95% of all the values of x̄ will fall within 2 standard deviations. Since the standard deviation of the sampling distribution of x̄ is 1.897, margin of error = 2(1.897) = 3.794.The correct answer is: 3.794

Front 148

True or false: We obtain the margin of error for estimating μ by using facts about the sampling distribution of x̄.

Back 148

true

Front 149

If x̄ = 10.2 and margin of error = 0.5, what is the confidence interval estimate for μ?

Back 149

The confidence interval is x̄– m, x̄ + m or (10.2 – 0.5, 10.5 + 0.5) = (9.7, 10.7)The correct answer is: (9.7, 10.7)

Front 150

Which of the following are the three simple conditions for estimating μ with the interval: ( x̄ – z*(σ / √n), x̄ + z*(σ / √n) )?

Back 150

For this formula, the data must be from an SRS, σ must be known, and the sampling distribution of x̄ must be either Normal (which happens when the population distrubtion is Normal) or approximately Normal (which happens when the sample is large and random and we can apply the Central Limit Theorem).The correct answer is: I, IV, V

Front 151

An administrator in a very large company wants to estimate the mean level of nitrogen oxides (NOX) emitted in the exhaust of a particular car model in their very large fleet of cars. Historically, nitrogen oxide levels have been known to be Normally distributed with a standard deviation of 0.15 g/ml. The administrator decides to obtain a 98% confidence interval estimate for the mean nitrogen oxide level of all cars of a particular model in the fleet. What is the parameter that the administrator wants to estimate?

Back 151

We want to estimate the mean of the population. The population is all cars of a particular model in their very large fleet of cars and the response variable is the level of nitrogen oxides emitted in the exhaust. So we want to estimate the mean level of nitrogen oxides (NOX) emitted in the exhaust of all cars of a particular model in their very large fleet of cars.The correct answer is: The mean level of nitrogen oxide of all cars of a particular model in the very large fleet.

Front 152

A Gallup Poll found that 51% of the people in its sample said "Yes" when asked, "Would you like to lose weight?" Gallup announced: "With 95% confidence for results based on the total sample of national adults, one can say that the margin of sampling error is ±3%."What is the 95% confidence interval estimate for the percent of all adults who want to lose weight?

Back 152

A confidence interval is computed using "estimate ± margin of error". The estimate is 51% and the margin of error is 3%. So, the confidence interval is 51% ± 3% or (48%, 54%).The correct answer is: (48%, 54%)

Front 153

A correct interpretation of confidence needs to address the following two points: (1) Confidence involves intervals from repeated samples and (2) 95% of these intervals contain the parameter being estimated. The only answer that does this is the answer "95% of the confidence intervals computed from the results of all possible samples will contain the percentage of all national adults who want to lose weight (i.e., the value of the parameter)." Saying, "computed from the results from all possible samples" fills the first requirement and saying, "95% of the confidence intervals . . .

Back 153

will contain the percentage of all adults who want to lose weight . . ." fills the second requirement. The correct answer is: 95% of the confidence intervals computed from the results of all possible samples will contain the true percentage of all national adults who want to lose weight.

Front 154

Fill in the blank: In confidence interval estimation we use a confidence interval to estimate...blank

Back 154

the value of a population parameter

The purpose of confidence intervals is to estimate the value of a population parameter using an interval. Confidence is an adjective to remind us that we have a certain degree of confidence in our interval and that we are not 100% certain about the estimate.The correct answer is: the value of a population parameter

Front 155

Confidence intervals estimate parameters. They do not tell us where the data are. In other words, a confidence interval estimate gives a range of reasonable values for μ, but does NOT give a range for most of the data.

Back 155

A confidence interval usually only contains a small portion of the data.

Front 156

An administrator in a very large company wanted to estimate the mean level of nitrogen oxides (NOX) emitted in the exhaust of a particular car model in their very large fleet of cars. Suppose the correct computation of the 98% confidence interval estimate is (0.87 g/ml, 0.97 g/ml). Which of the following is a correct interpretation of this 98% confidence interval?

Back 156

We are 98% confident that the value of the mean nitrogen oxide level, μ, for all cars of a particular model is between 0.87 g/ml and 0.97 g/ml.

The parameter to be estimated is the mean level of nitrogen oxide of all cars of a particular model in the very large fleet. So the correct interpretation is "We are 98% confident that the mean nitrogen oxide level of all cars of a particular model in the very large fleet is somewhere between 0.87 and 0.97. This is a good example of an exam question on interpreting confidence intervals.

Front 157

Refer to the 98% confidence interval estimate of (0.87 g/ml, 0.97 g/ml) given in the above question. Which one of the following is a correct interpretation of 98% confidence?

Back 157

98% of the time, using the same procedure as we used to obtain the interval (0.87 g/ml and 0.97 g/ml), we will obtain confidence intervals that contain the value of μ.

We state confidence in terms of the procedure not a specific calculated interval like (0.87 g/ml, 0.97 g/ml). This is because we know that 98% of all possible samples yield 98% confidence intervals that contain the value of μ. Note that the correct answer says "confidence intervals" in plural not singular.

Front 158

Suppose the 95% confidence interval estimate for the mean monthly cost for Internet service for all Internet users is ($19.90, $21.90). Which of the following is a correct interpretation of this 95% confidence interval?

Back 158

We are 95% confident that the mean monthly cost for Internet service paid by all Internet users is between $19.90 and $21.90.

The parameter to be estimated is the mean monthly cost for Internet service paid by all Internet users. So the correct interpretation is "We are 95% confident that the mean monthly cost for Internet service paid by all Internet users is somewhere between $19.90 and $21.90." This is a good example of an exam question on interpreting confidence intervals.

Front 159

Consider the following interpretation of 90% confidence: "The probability that the mean of all axle diameters lies somewhere in the interval (0.98 centimeters, 1.02 centimeters) is 0.90." Is this interpretation of 90% confidence correct or incorrect? Why or why not?

Back 159

Incorrect because it states "probability on one specific calculated interval" rather than "confidence in the procedure."

Since the interval has been calculated, we know that the sample has already been taken. Because of this, we cannot state a probability on the value of the parameter being somewhere in the specific calculated interval other than by saying it is "0.0 or 1.0--we just don't know which."

Front 160

Consider the following interpretation of 90% confidence: "90% of all possible confidence intervals computed using the same procedure used to obtain (0.98, 1.02) will contain the value of μ." Is this interpretation of 90% confidence correct or incorrect? Why or why not? Note: Interpreting "confidence level" is not the same as interpreting a confidence interval.

Back 160

Correct because it is stated in terms of the confidence interval procedure not one specific calculated interval.

This interpretation of 90% confidence refers to the confidence interval procedure and not just to the one specific calculated interval (0.98, 1.02). We express confidence as the percentage of the time the procedure produces intervals containing the value of μ, not as the percentage of the time one specific calculated interval (0.98, 1.02) contains μ when we interpret 90% confidence.

Front 161

Suppose the confidence interval turned out to be (8.001 , 13.829). What is an appropriate interpretation of this interval?

Back 161

We are 99% confident that the true mean number of prayers said in the BYU Testing Center by all BYU students is between 8.001 and 13.829 prayers. Correct

A correct interpretation of a confidence interval gives 1) a statement of confidence which in this case is 99%, 2) a statement of the parameter which is "the true mean number of prayers said in the BYU testing center by BYU students" and 3) the interval.

Front 162

Fill in the blank: An observed effect that is due to chance variation will have a ___________ probability.

Back 162

large (> 0.05)

A "large" observed effect will have a small (< 0.05) probability, but an observed effect whose size is "due to chance" will have a large (> 0.05) probability.

Front 163

True or false: An outcome that would rarely happen if a claim were true is good evidence that the claim is not true.

Back 163

True

This is the basic reasoning of a test of significance.

Front 164

In planning a study of the birth weights of babies whose mothers did not see a doctor before delivery, a researcher states the hypotheses asH0: x̄ = 1000 gramsHa: x̄ < 1000 gramsWhat's wrong with this?

Back 164

The hypotheses must be stated in terms of population parameters such as μ.

Using x̄ symbols or values in hypotheses will always be wrong. Parameter symbols and hypothesized parameter values must always be used in hypotheses. For hypotheses in Chapter 15, always use μ.

Front 165

True or false: The alternative hypothesis, Ha is a statement of status quo or no difference.

Back 165

The null hypothesis, H0 is a statement of status quo or no effect.The correct answer is: False

Front 166

According to the Environmental Protection Agency (EPA), the Honda Civic hybrid car gets 51 miles per gallon (mpg) on the highway. The EPA ratings often overstate true fuel economy. Larry keeps careful records of the gas mileage of his new Civic hybrid for 3000 miles of highway driving. His result is x̄ = 47.2 mpg. Larry wonders whether the data show that his true long-term average highway mileage is less than 51 mpg. What are his null and alternative hypotheses?

Back 166

Since Larry wants to see whether that the mean mpg is less than 51 mpg, he wants to show that μ < 51. This is will be the alternative hypothesis or Ha. We put equal in the null hypothesis so H0 is μ = 51. Using x̄ in the hypotheses will always be wrong; similarly, using x̄ values such as 47.2 will always be wrong.The correct answer is: H0: μ = 51 and Ha: μ < 51

Front 167

True or false: The closer the observed value for x̄ is to μ0, the smaller the P-value (i.e., the smaller the observed effect, the smaller the P-value.)

Back 167

Actually, the opposite is true. The farther the observed value for x̄ is from μ0, the smaller the P-value. This is because the tail area of the sampling distribution of x̄ gets smaller as x̄ gets farther from μ0.The correct answer is: False

Front 168

True or false: We always assume H0 is true when we compute P-value.

Back 168

True

In order to compute P-value, we must first compute the test statistic or z-score. The test statistic requires a numerical or claimed value for μ. This value is μ0 as specified in H0. Thus, by using μ0 in the formula of the test statistic, we are assuming H0 is true.

Front 169

The weight of a carton of a dozen eggs produced by a certain breed of hens is supposed to be 780 grams. A quality manager checks five cartons of eggs (n = 5) to see whether the mean weight differs from 780 grams. What hypotheses does the quality manager wish to test?

Back 169

The correct answer is: H0: μ = 780 and Ha: μ ≠ 780

Front 170

True or false: P-value is the probability that H0 is true.

Back 170

false

This false statement is a misconception that many students have about P-value, but P-value is NOT the probability that H0 is true. P-value is a conditional probability---conditional on H0 being true. P-value is a probability on a statistic ASSUMING H0 is true as opposed to a probability THAT H0 is true. The "assuming H0 is true" part defines the center of the sampling distribution of x̄. Once this center is defined, we can compute the probability of getting a statistic (an x̄) as extreme or more extreme than the value we observed for x̄, i.e., we can compute P-value. In short, P-value is the probability on a statistic if H0 is true.

Front 171

Suppose we are testing H0: μ = 40 versus Ha: μ > 40. If α = 0.01 and P-value = 0.08, what should we conclude?

Back 171

Since P-value > α, we fail to reject H0 and say we have insufficient evidence to believe the mean is greater than 40.The correct answer is: We say that we have insufficient evidence to conclude that the population mean is greater than 40.

Front 172

True or false: A small P-value gives evidence for H0.

Back 172

The opposite is true--a small P-value gives evidence against H0; i.e., evidence that H0 is incorrect and that Ha is correct. We reject H0 when the P-value is less than α.The correct answer is: False

Front 173

Suppose we are testing H0: μ = 40 versus Ha: μ > 40. If α = 0.10 and P-value = 0.08, what should we conclude?

Back 173

Since P-value < α, we reject H0 and conclude that the population mean is greater than 40 (i.e., believe Ha).The correct answer is: We conclude that the population mean is significantly greater than 40.

Front 174

In the SOLVE step of the procedure we must first check the conditions:Randomization, Normality of the Sampling Distribution of x̄, and σ Known.Are these conditions met? Why or why not?

Back 174

Randomization is met because a random sample was taken from chocolate chip cookies made by this brand. Normality of the Sampling Distribution of x̄ is met because it is given that calorie amount is Normally distributed. If the Population is normally distributed the Sampling distribution of x̄ will be Normally distributed. σ Known is met as we have σ = 24.12.

Front 175

The SOLVE step of the procedure yielded an x̄ = 120.35714. What is the proper z-statistic for this test?

Back 175

From the formula on the formula sheet on the back of the syllabus we see z = (x̄ - μ) / [σ / √(n)] = (120.357 - 100) / [24.12/√(14)] = 20.357 / 6.44643 = 3.16. So the proper z-statistics for this test is z = 3.16.The correct answer is: z = 3.16

Front 176

As part of the SOLVE step, suppose the z-statistic was 3.22, what would the proper p-value be for this test?

Back 176

The proper p-value for the test with hypotheses , is 0.0006.The correct answer is: 0.0006

Front 177

If the p-value for the test was 0.0042 (it's not) what would you conclude at the α = 0.05 significance level, in context?

Back 177

Since the p-value for this test (0.0042) was less than the 5% significance level, we have sufficient evidence to conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.The correct answer is: Since the p-value for this test (0.0042) was less than the 5% significance level, we have sufficient evidence to reject the null hypothesis and conclude that the true mean calorie amount of all 1-ounce chocolate chip cookies produced by this brand is greater than 100 calories.

Front 178

When do you reject H0 and conclude that Ha is correct?

Back 178

Whenever P-value < α.

P-value < α tells us that observed effect would rarely happen when the null hypothesis is true. That gives good evidence that the null hypothesis is not true so we reject it. Note that “P-value <α,” “statistically significant,” and “reject H0” are equivalent.

Front 179

Suppose you are testing H0: μ = 27 versus Ha: μ > 27 and the P-value is 0.0218. What can you conclude at α = 0.05?

Back 179

Since P-value = 0.0218 < 0.05 = α, we have evidence to reject H0. So we say, “We conclude that Ha is correct.” In this case we say, “We have sufficient evidence to conclude that the mean is significantly greater than 27.”The correct answer is: The mean is significantly greater than 27.

Front 180

Suppose you are testing H0: μ = 76 versus Ha: μ ≠ 76 and the P-value is 0.019. What can you conclude at α = 0.02?

Back 180

Since P-value = 0.019 < 0.02 = α, we have evidence to reject H0. So we say, “We conclude that Ha is correct.” In this case we say, “We have sufficient evidence to conclude that the mean is significantly different from 76.” Note: We do not re-double P-value for this two-sided test. The tail area was already doubled to obtain this P-value.The correct answer is: The mean differs significantly from 76.

Front 181

Suppose you are testing H0: μ = 23.6 versus Ha: μ > 23.6 and the test statistic is z = 2.03. What is the P-value for this test?

Back 181

Since this is an upper tailed test, we need to find the area greater than z = 2.03. When we look up z = 2.03 in the standard Normal table, we get 0.9788. This is the area less than z = 2.03. To find the area greater than z = 2.03, we compute P-value = 1 – 0.9788 = 0.0212. (Note: Since the standard Normal curve is symmetric, we could also look up z = –2.03 to get 0.0212 as the area greater than z = 2.03.)The correct answer is: 0.0212

Front 182

Suppose you are testing H0: μ = 5 versus Ha: μ ≠ 5 and the test statistic is z = –0.81. What is the P-value for this test?

Back 182

This is a two-sided test, so we need to find the tail area for both tails. We first need to find the area less than z = –0.81. When we look up z = –0.81 in the standard Normal table, we get 0.2090. This is the area less than z = –0.81. Because this is a two-sided test, we must double the tail area to get P-value = 2(0.2090) = 0.4180.The correct answer is: 0.4180

Front 183

Nitrogen oxide levels are not to exceed 0.9 grams per mile (g/ml). An administrator in a very large company is assigned to test whether the mean level of nitrogen oxides (NOX) emitted in the exhaust of a particular car model in their very large fleet of cars exceeds 0.9 grams per mile. He randomly samples 40 cars and finds x̄= 0.97. If σ = .15 g/ml grams, what is the value of the standardized z test statistic or z-score?

Back 183

The correct answer is: 2.95

Front 184

When asked to explain the meaning of "the P-value was P = 0.03," a student says, "This means there is only a 0.03 probability that the null hypothesis is true." Is this a correct explanation?

Back 184

The problem is that P-value is a probability of observing the data we did in a conditional situation. The conditional situation is that the null hypothesis is true. So P-value is a probability IF the null is true, not THAT the null is true. P-value is the probability of getting a test statistic as extreme or more extreme than observed IF (or assuming) the null hypothesis is true.The correct answer is: No, this is a false explanation. A P-value is the probability, assuming the null hypothesis is true, that the test statistic will take a value at least as extreme as that actually observed.

Front 185

True or False: The parameter of interest is the mean change in SAT mathematics exam score (second try minus first try) of the random sample of 46 high school students.

Back 185

The statement is FALSE. It should be stated: μ for this scenario is the mean change in SAT mathematics exam score (second try minus first try) of all high school students.

Front 186

SOLVE step:Can we say that the condition of "Normality of the Sampling Distribution of x̄" is met? Why or why not?

Back 186

Yes, because the sample size is large enough to apply the Central Limit Theorem.

Front 187

The mean of the change in scores is x̄ = 13.108696. Using this, find the P-value for this test.

Back 187

We first need to compute the z test statistic is z = (x̄ - μ0) / [σ /√(n)] = (13.108696-0.0) / [50/√(46)] = 1.7781 or 1.78. Looking this z-score up in the standard Normal table, we get a probability of 0.9625. But this is the cumulative probability and we need tail probability for P-value. Subtracting from 1.0 we get P-value = 0.0375.

Front 188

CONCLUDE step:Suppose the P-value is 0.0271. What should we conclude? Use α = 0.01.

Back 188

Since P-value = 0.0271 > 0.01 = α, we should fail to reject H0 and conclude that we have insufficient evidence to say that the mean change in the SAT mathematics exam is greater than 0.The correct answer is: We should fail to reject the null hypothesis and say that we have insufficient evidence to conclude that the mean change in the SAT mathematics exam is greater than 0.

Front 189

1

What is the purpose of a 95% confidence interval for μ?

Back 189

A 95% confidence interval is an interval estimate for μ with associated 95% confidence. It gives an interval of reasonable values for μ.The correct answer is: To give a range of plausible values for μ.

Front 190

True or false: A 90% confidence interval can be used to perform a two-sided test on H0: μ = μ0 versus Ha: μ ≠ μ0 at α = 0.10.

Back 190

This true statement has two important components. First, the 90% confidence level matches with α = 0.10. Second, the test needs to be two-sided. Because both of these components are met, the statement is true.The correct answer is: True

Front 191

A 90% confidence interval for μ is given as (11.2, 14.7). On the basis of this interval, can we reject H0: μ = 15 (versus Ha: μ ≠ 15) at α = 0.10?

Back 191

A confidence interval for m gives reasonable values for μ. Since the value 15 is NOT between 11.2 and 14.7, it is not a reasonable for μ and we can reject H0: μ = 15.The correct answer is: Yes

Front 192

A researcher is interested in estimating the mean yield (in bushels per acre) of a variety of corn. From her sample, she calculates the following 95% confidence interval for the mean yield: (118.74, 128.86). Her colleague wants to test whether the mean yield for the population differs from 120 bushels per acre. (H0: μ = 120 versus Ha: μ ≠ 120). On the basis of this confidence interval, what can her colleague conclude at α = 0.05?

Back 192

Her colleague cannot reject the null hypothesis and therefore, cannot conclude that the mean yield differs significantly from 120.

Since 120 is between 118.74 and 128.86, 120 is a reasonable value for μ based on these data. Therefore, her colleague cannot reject H0 and cannot conclude that the mean yield differs significantly from 120. Also, she cannot conclude that μ is equal to 120 because the 95% confidence interval tells her the possible values for μ based on x¯ = 797 grams, that the value of μ is somewhere between 118.74 and 128.86 with 95% confidence. So the value of μ could be 120, but it could also be any other value in this interval based on these data with 95% confidence.

Front 193

A New York Times poll on women's issues interviewed 1025 women randomly selected from the United States, excluding Alaska and Hawaii. The poll found that 47% of the women said they do not get enough time for themselves. The poll announced a margin of error ± 3% for 95% confidence in its conclusions. What does this margin of error account for?

Back 193

sampling variation

Margin of error accounts for only sampling variability that results from taking random samples.The correct answer is: Sampling variation

Front 194

Which of the following is the most important condition for inference in practice?

Back 194

Whether data collection was randomized.

The most important condition in inference is checking to make sure that the data were appropriately collected. The second most important condition is checking either the population distribution or the sampling distribution of the sample statistic for Normality.

Front 195

True or false: One should always check for outliers before performing inference.

Back 195

true

Outliers suggest that the data are from a non-Normal population, so inference that requires Normality should not be performed. That's why checking for outliers is so important.The correct answer is: True

Front 196

From the actions listed below, which must we do first in the SOLVE step when performing inference?

Check conditions Correctb. Compute the test statistic and find P-valuec. Draw conclusions

Back 196

check conditions

The number we get for P-value will not be correct if the conditions are not met. So before we compute the test statistic and P-value, we must check the conditions so that the number we get for P-value will be correct.

Front 197

A marketing consultant is studying shopping habits of shoppers at a particular supermarket. They observe 50 consecutive shoppers at the supermarket and record how much each shopper spends in the store. Which one of the following is NOT a reason that it may be risky to act as if 50 consecutive shoppers at a particular time are an SRS of all shoppers at this store.

Back 197

Some people don't shop at that supermarket.

Front 198

True or false: x̄ computed using results from a simple random sample can appropriately be used in this confidence interval formula for μ : x̄ ± z*(σ / √n).

Back 198

true

Front 199

Fill in the blank: For fixed sample size, increasing level of confidence __________________ margin of error.

Back 199

increases

The formula for margin of error is z*σn√. As level of confidence increases, z* increases, and consequently, margin of error also increases.

Front 200

True or false: For a given level of confidence, the larger the sample size, the wider the resulting confidence interval.

Back 200

false

The formula for margin of error is z*σn√. Since sample size is in the denominator, increasing sample size decreases margin of error. Thus, the smaller the margin of error, the narrower the resulting confidence interval.

Front 201

Fill in the blank: For a given level of confidence and standard deviation, decreasing sample size __________________ margin of error.

Back 201

increases

The formula for margin of error is z*σn√. Hence, increasing sample size decreases margin of error since it is in the denominator. This was demonstrated by questions 11 and 12 with margin of error = 3.099 for n = 1000 and margin of error= 1.550 for n = 4000. Also, according to the Law of Large Numbers, x¯ gets closer to μ as the sample size increases. So margin of error has to decrease as sample size increases.

Front 202

In a national poll, Gallup asked the question, "Do you feel confident that you are safe in your community?" Gallup announced the poll's margin of error for 95% confidence as +/- 3 percentage points. Which of the following sources of error are included in this margin of error?

Back 202

There is chance variation in the random selection of the telephone numbers dialed.

Front 203

True or false: Population size does not affect margin of error.

Back 203

True

The formula for margin of error is z*σn√. Sample size, population standard deviation and level of confidence affect margin of error, but the size of the population is not part of the formula.

Front 204

1

A company compares two cereal box designs for a flaky wheat cereal with marshmallows by placing both boxes with the different designs in random order on the shelves of several markets. Checkout scanner data on more than 5000 boxes bought show that more shoppers bought Design A than Design B. The difference is statistically significant (P-value = 0.02).The hypotheses are: H0: Customers have no preference of Design A over Design B versus Ha: Customers prefer Design A over Design B. Since the difference is statistically significant, can we conclude that consumers prefer Design A over Design B? Why or why not?

Back 204

First of all, this is an experiment. Further, since P-value = 0.02 < α = 0.05, the results are statistically significant and we can reject the null hypothesis. Thus, we can conclude that customers prefer Design A over Design B.The correct answer is: Yes, because this is an experiment and for a P-value of 0.02 we reject the null hypothesis at α = 0.05.

Front 205

If the company is currently using Design B, should it switch to Design A since the null hypothesis was rejected? Be sure to consider sample size and the issue of practical significance.

Back 205

Since the sample size of 5000 was so large, the company has to consider whether the switch is cost effective. It may be that switching will cost more than the added revenue. If this is the case, the company shouldn’t switch even if the results are statistically significant. However, if switching results in higher revenue even after the cost associated with the switch, then the company should switch.The correct answer is: Yes, because the null hypothesis was rejected, but only if the added revenue from switching the package design outweighs the costs associated with the switch.

Front 206

True or false: Statistically significant results are also practically significant.

Back 206

false

Statistical significance is determined by comparing P-value with α Practical significance is determined by subjectively assessing the observed effect. (Note: The observed effect is the numerator of the standardized test statistic or the difference between the observed statistic value and the claimed parameter value.)

Front 207

Many people are worried about getting cancer. In one study on causes of cancer, researchers performed tests of significance on all small appliances on the market. Usage of two of these, hair dryers and small black and white TV's, was found to be associated with cancer. Can causation be established from this study?

Back 207

This study is observational because researchers cannot ethically try to cause people to get cancer. Consequently, causation cannot be established.The correct answer is: No, this study was observational so causation cannot be established.

Front 208

True or false: Referring to the study described in question 4, when performing tests of significance on association between cancer and appliance usage for all small appliances (at least 50 small appliances), we expect to find two or three significant due to chance at α = 0.05.

Back 208

true

At α = 0.05, if we test 50 small appliances, we expect to find α times n = (0.05) (50) = 2.5 appliances associated with cancer just due to chance. So finding two to be significant is expected, not a surprise. The significant relationship between usage and cancer for two small appliances, hair driers and black and white TV's, is probably meaningless.

Front 209

A 95% confidence interval for a population mean, μ, is 31.5 ± 3.5. On the basis of this interval, should you reject the null hypothesis that μ = 34 at the 5% significance level? On the basis of this interval, should you reject the null hypothesis that μ = 36 at the 5% significance level? Assume both are two-sided tests.

Back 209

The confidence interval is 31.5 – 3.5 = 28.0 to 31.5 + 3.5 = 35.0 or (28.0, 35.0). Since the value of 34 is contained in this interval, we cannot reject the null hypothesis that μ = 34. But the value of 36 is NOT contained in this interval, so we CAN reject the null hypothesis that μ = 36.The correct answer is: We cannot reject the null hypothesis that μ = 34, but can reject the null hypothesis that μ = 36.

Front 210

"How much do you plan to spend for gifts this Valentine's Day?" An interviewer asks this question of 250 customers at a medium-sized shopping mall. The sample mean of the responses is x̄ = $37. Assume that σ = $6.5. A 99% confidence interval for the mean gift spending of all adults was computed to be ($35.941, $38.059). This confidence interval can't be trusted to give information about the spending plans of all adults. Why not?

Back 210

because the chosen sample is not an simple random sample

The sample consisted of customers contacted at a large shopping mall. Since this is a sample of convenience not an SRS, the results of the confidence interval cannot be trusted. Note: The sample size is large enough to apply the Central Limit Theorem so the sampling distribution of x̄ would be approximately Normal IF the sample were randomly selected.

Front 211

In the absence of special preparation SAT mathematics (SATM) scores in recent years have varied Normally with mean μ = 475 and σ = 100. One hundred students go through a rigorous training program designed to raise their SATM scores by improving their mathematics skills. Carry out a hypothesis test of H0: μ = 475 Ha: μ > 475in each of the following situations:Suppose that the average score for 100 students is x¯ = 491.4. Is this result statistically significant at the 5% level? To answer, compute z-score and find P-value.

Back 211

Looking this z-score up in the standard Normal table, we get a probability of 0.9495. To get area on the right we subtract this probability from 1.0 to get 0.0505. Since P-value = 0.0505 > α = 0.05, we fail to reject the null hypothesis and declare the result to be NOT statistically significant.The correct answer is: No

Front 212

Suppose that the average score for 100 students is x̄ = 491.5. Is this result statistically significant at the 5% level?

Back 212

The z test statistic is z = [x̄ - μ0] / [(σ / √n)] = [491.5-475] / [(100 / √100)] = 1.650. Looking this z-score up in the standard Normal table, we get a probability of 0.0495. Since P-value = 0.0495 < α = 0.05, we reject the null hypothesis and declare the result to be statistically significant.The correct answer is: Yes

Front 213

When performing a test of significance, when should we select α?

Back 213

When defining the research problem in the PLAN step

Part of setting up and defining the problem in the PLAN step is stating the hypotheses and selecting α.The correct answer is: When defining the research problem in the PLAN step.

Front 214

True or false: α should always be chosen before a P-value is computed.

Back 214

true

The value of α should always be selected in the PLAN step when defining the problem.The correct answer is: True

Front 215

City officials in Provo would like to estimate the average income of apartment dwellers with a margin of error of $500 using a 98% confidence level. If the population standard deviation is known to be σ = $7000, how many apartment dwellers should they sample?

Back 215

1060.414 so we should use an n of 1061. We round up because sample size is a whole number. (You should always round up when computing sample size.)The correct answer is: 1061

Front 216

Referring to the above question, suppose the city officials decided that they should estimate μ with 99% confidence and not 98%. What effect would this have on the required sample size, n?

Back 216

By increasing the confidence level, the value for z* will increase and therefore, so will the required sample size increase.The correct answer is: It would increase the required sample siz

Front 217

Referring to the above two questions, suppose the city officials decided they needed to estimate the mean salary with margin of error of $1000 rather than $500. What effect would this have on the required sample size, n?

Back 217

By allowing for a bigger margin of error, the executives will need to sample fewer apartment dwellers.The correct answer is: It would decrease the required sample size.

Front 218

1

Which two of the following are correct decisions?I. Rejecting H0 when H0 is correct.II. Rejecting H0 when H0 is wrong.III. Failing to reject H0 when H0 is correct.IV. Failing to reject H0 when H0 is wrong.

Back 218

We want to reject H0 when it is false (wrong) and we don’t want to reject H0 when it is true (correct).The correct answer is: II, III

Front 219

A paramedic comes upon the scene of an accident and tests the hypothesesH0 : victim is alive versusHa: victim is dead.What would be his/her type I error?

Back 219

Believing victim is dead when victim is really alive.

A type I error is to reject H0 (i.e., believe Ha) when H0 is really correct. In this case rejecting H0 is to reject that the victim is alive and believe that victim is dead. But H0 is true—the victim is really alive. So, the type I error is for the paramedic to believe that the victim is dead when the victim is really alive.

Front 220

Referring to the above question, which of the following errors is most serious?

Back 220

If an accident victim is alive and doesn’t get help soon, that victim will probably die. However, if the victim is dead and the paramedic works on him or her, the paramedic may waste time, but hopefully doesn’t neglect another victim’s needs and cause that victim to die.The correct answer is: Believing victim is dead when victim is really alive.

Front 221

In general, if the error you considered most serious is the Type I Error, what should you set α as?

Back 221

is the probability of a Type I Error. If we consider the Type I Error to be the most serious, we want to set α as small as possible. The correct answer is: 0.0001 which makes the Type I Error probability low

Front 222

The CEO of a large company feels that redesigning the production line will increase output. She will test the hypotheses:H0: Redesigning will not increase output.Ha: Redesigning will increase output.If she finds evidence to reject H0, she will initiate an expensive redesign of the production line. What error will she make if she rejects H0 and initiates an expensive redesign when redesigning will not increase output?

Back 222

A type I error occurs by rejecting H0 when it is true. By rejecting H0 and initiating the redesign when redesigning will not increase output, she makes a type I error.The correct answer is: Type I error

Front 223

Which error would the CEO make by failing to reject H0 and failing to redesign the production line when redesigning would really increase output?

Back 223

The type II error is made by failing to reject H0 when H0 is wrong (false). In other words, a type II error is made if you believe H0 when Ha is correct. So failing to initiate the design when redesigning would increase output is a type II error.The correct answer is: Type II error

Front 224

Referring to question 5, what are the consequences of initiating an expensive redesign when redesigning will not increase output?Select one:a. Company pays for expensive redesign without recouping the cost with increased output.b. Company does not get the increased output that it could have had.

Back 224

Company pays for expensive redesign without recouping the cost with increased output.

Redesigning when it costs money, but will not increase output will cost the company without the company being able to recoup the cost.

Front 225

If we set α at 0.05, what can we do to increase power?Select one:a. Increase sample sizeb. Decrease sample sizec. Increase the probability of a type II error

Back 225

Increase sample size

Increasing the sample size makes both sampling distribution curves get taller and skinnier. This separates the two curves as displayed in question 8 above and effectively increases power.The correct answer is: Increase sample size

Front 226

Your company markets a computerized medical diagnostic program used to evaluate thousands of people. The program scans the results of routine medical tests (pulse rate, blood tests, etc.) and refers the case to a doctor if there is evidence of a medical problem. The program makes a decision about each person. The hypotheses are:H0: The patient is healthy.Ha: The patient has a medical problem.Which of the following describes a type I error in this situation? Select one:

a. Discharging a patient with a medical problemb. Sending a patient with a medical problem to the doctor.

c. Sending a healthy patient to the doctor.

d. Discharging a healthy patient.

Back 226

Sending a healthy patient to the doctor.

A type I error is made when a correct null hypothesis is rejected. In this case, if the program rejects a correct null hypothesis, it decides a patient has a medical problem when that patient is actually healthy. The consequence of this decision is that a healthy patient gets sent to the doctor.

Front 227

Which of the following describes a type II error in this situation?

Back 227

Discharging a patient with a medical problem.

A type II error is made when an incorrect null hypothesis is NOT rejected. In this case, if the program fails to reject an incorrect null hypothesis, it decides a patient is healthy when that patient actually has a medical problem and discharges that patient.

Front 228

Suppose you are a patient most concerned about being discharged if you were sick. Which error probability would you choose to make smaller?Select one:

a. Type I error probability

b. Type II error probability

Back 228

Type II error probability

If you are most concerned about being discharged if you were sick (i.e., failing to reject the null when it is incorrect), you are most concerned about making a type II error so you would want the type II error probability to be smallest.The correct answer is: Type II error probability

Front 229

Suppose you work for an insurance company which covers these types of routine doctor visits for some of its policyholders. The company desires to minimize costs by not sending a healthy patient to a doctor. Which error probability would you choose to make smaller?

Select one:

a. Type I error probability

b. Type II error probability

Back 229

Type I error probability

If you are most concerned about sending a healthy patient to the doctor (i.e., incorrectly rejecting the null when it is correct), you are most concerned about making a type I error so you would want the type I error probability to be smallest.The correct answer is: Type I error probability

Front 230

Which of the following questions does a test of significance answer?

Select one:

a. Is the sample or experiment properly designed?

b. Is the observed effect due to chance alone?

c. Is the observed effect important?

d. Is the sample size large enough?

Back 230

Is the observed effect due to chance alone?

A test of significance answers the question, “ Is the observed effect due to chance?” We ask the question, “Is the sample or experiment properly designed?” when assessing data collection. We ask the question “Is the observed effect important?” when determining practical significance. And sometimes we ask, “Is the sample size large enough? when we fail to reject the null hypothesis and feel we may have made a type II error. So it’s very important for you to know when to ask each question.The correct answer is: Is the observed effect due to chance alone?