St 351 Midterm 2 And Final

Front 1

trial

Back 1

one repetition of a experimental situation

Front 2

outcome

Back 2

what happens in the trial

Front 3

relative frequency

Back 3

# times an outcome of interest occurs/total # of trials

Front 4

regularity

Back 4

idea that in the long run that there will be a set proportion to outcomes

Front 5

law of large numbers

Back 5

long run relative frequency of repeated independent trials will get closer to the true relative frequency as the number of trials increases

Front 6

sample space

Back 6

list of all possible outcomes in a probability experiment

Front 7

event

Back 7

outcome of collection of outcomes in a probability experiment

ex: A= HH (getting heads then heads again for flipping a coin)

Front 8

equally likely outcomes

Back 8

have the same probability of occuring

Front 9

probability

Back 9

# outcomes of an event/# of total outcomes

Front 10

rules of probability

Back 10

P is always between 0 and 1

probabilities of all possible outcomes sum to 1

Front 11

complement rule

Back 11

P(B) = 1 - P(A)

Front 12

Addition rule

Back 12

If two events are mutually exclusive then P(A or B) = P(A) + P (B)

Front 13

mutually exclusive

Back 13

no outcomes in common

Front 14

mutliplication rule

Back 14

if two events are independent of each other then P (A and B) = p(A) x P(B)

Front 15

indepedent

Back 15

the outcome of A does not depend on the outcome of B

Front 16

disjoint

Back 16

when two events have nothing in common

Front 17

what is one way to tell that events are independent

Back 17

see if selection of one event affects the probability of the other

OR

The probability of (A or B) = P(A) * P(B)

Front 18

General addition rule

Back 18

P(A or B) = P (A) + P (B) - P (A and B)

Front 19

what does the word and imply

Back 19

that they events occur at the same time

Front 20

what does the word or imply

Back 20

that event A occurs, event B occurs, or event A and B occurs at the same time

Front 21

conditional probability

Back 21

P(B|A) = P (A and B)/P(A)

Front 22

How is P(B|A) read

Back 22

the probability of event B given that event A has already occured

Front 23

General multiplication rule

Back 23

P(A and B) = P(A) x P(B|A)

Front 24

what does the box in a venn diagram represent

Back 24

the sample space

Front 25

what does a circle in a venn diagram represent

Back 25

an individual event

Front 26

If events in a venn diagram do not overlap what does that imply

Back 26

that the events are disjoint

Front 27

premise of conditional probability

Back 27

finding the probability of one event given that a second event has already occured

Front 28

What does an event to the power C represent

Back 28

the compliment

ex: if there are two events blue marbles and red marbles. If A is choosing a red marble, then Ac is chosing a blue marble

Front 29

what does truly random mean

Back 29

that every outcome is equally likely to happen

Front 30

random variable

Back 30

a variable whose value is a numerical outcome of some random event

for example the number of heads that occur during 10 coin flips is a random variable because it is subject to a random event

Front 31

what is a typical notation of random vairable

Back 31

X or Y

Front 32

discrete random variable

Back 32

finite number of possible outcomes for an event

Front 33

continous random variable

Back 33

can contain any value within a range of values

Front 34

what does a probability distribution of X (a random variable) mean?

Back 34

it is the range of values that the random variable probability lies within

Front 35

how is the probability of a continuous random variable described

Back 35

by a density curve

Front 36

How is probability determined by a density curve?

Back 36

It is the area underneath the curve between the values in which the continuous variable falls

Front 37

how does one distinguish between a variable of interest and a random variable

Back 37

variable of interest represents the piece of information one wishes to receive from just one observation. (ex: what happens when we flip a coin? we get a heads or tail)

random variable: combines all the observations together. Keeping track of the number of times an outcome occurs. (ex: the number of times a coin flip comes up heads)

Front 38

What is a nuance if the variable of interest is quantitative

Back 38

the random variable will be the same as well

Front 39

If the variable of interest is categorical, the random variable will be

Back 39

discrete

Front 40

If the random variable is quantitative then the random variable is

Back 40

continuous

Front 41

when using density curves to calculate the probability under a curve, what is essential to remember

Back 41

that the area under the curve is 1

therefore l x w = 1

Front 42

on a continuous random variable density curve, what is the probability of a single value

Back 42

zero because it does not have a width on the graph

Front 43

what is he mean and standard deviation of a standard normal curve

Back 43

mean is 0 and standard deviation is 1

Front 44

z score

Back 44

is the number of standard deviation an observation is from the mean

Front 45

what does it mean if a z statistic is positive? what does it mean if it is negative?

Back 45

positive- greater than the mean
negative- less than the mean

Front 46

what does a z score of zero mean

Back 46

it is equal to the mean

Front 47

how does one calculate a z score

Back 47

observed value - mean / standard deviation

X -u / o

Front 48

if the population data are normally distrubutedd, then the sample means are

Back 48

normally distributed

Front 49

why use sample means

Back 49

because it is often difficult to get data for an entire population

Front 50

when are sample means normally distributed

Back 50

most of the time so long as the sample size for each sample is large enough

Front 51

what is always true about the population mean and the sample mean

Back 51

they are always equal

Front 52

what does ux mean? uxbar?

Back 52

ux = population mean
uxbar =the mean of the sample sample means

Front 53

what is always true about the standard deviation sof the population and sample mean

Back 53

the smaple mean will always be smaller

Front 54

equation for the standard deviation of the sample means

Back 54

o xbar = ox / srt(n)

n = sample size

Front 55

what is a condition of the standard deviation of sample means equation

Back 55

it is best when sampling is done with replacement

Front 56

sample mean

Back 56

is a mean of a collection of samples

often this is done many times in order to create a distribution of sample means

Front 57

alternative hypothesis notation

Back 57

Ha

Front 58

what is required about the null and alternative hypothesis

Back 58

they have the same hypothesized value

Front 59

what is often true about an alternative hypothesis

Back 59

it makes a statement where the population mean is greater or less than the hypothesized value or that it does not equal it

Front 60

What are conditions that lead to a one sided (one tailed) test

Back 60

when the alt hypothesis states that the population mean is greater or less than the hypothesized mean

Front 61

what are conditions that lead to a two sided (two tailed test)

Back 61

when the alt hypothesis states that the population mean does not equal the hypothesized value

Front 62

if there is a quantitative variable of interest, what information is needed to determine what statistical test should be done?

Back 62

n (sample size)
xbar (sample mean)
sx (sample standard deviation)
ox (population standard deviation)

Front 63

if the population standard deviation is known what stat test should be done?

Back 63

one sample t test

Front 64

if the sample standard deviationis known what stat test should be done

Back 64

one sample t test

Front 65

what are conditions of the one sample z test

Back 65

1) the sample is representative of the population
2) the distribution of sample means is normal
3) obervations are independent of each other

Front 66

what is the only way to guarantee that you have a representative sample from a population

Back 66

take a random sample

Front 67

what are assumptions of a one way z test

Back 67

that the sample is representative of the population

that the distrubution of sample means is approximately normal

observations are independent of each other

Front 68

what assumption is made for all hypothesis tests

Back 68

that observations are independent of each other

Front 69

p value

Back 69

the probability of getting a sample mean that is as or more extreme than the one that was observed during the experiment if the null hypothesis is true

Front 70

is it possible that a sample mean could have turned out to be a different value than the one hypothesized

Back 70

yes

Front 71

what action can be taken about the null hypothesis

Back 71

it can be rejected or fail to be rejected

Front 72

z statistic

Back 72

the number of standard deviations from what is observed from the mean

oberv-hypo/stand dev

Front 73

when do you reject the null hypothesis

Back 73

when it is small!

less than 0.1 is strong evidence
0.1 to 0.5 show suggestive evidence

0.05 to .1 weak evidence

.1 or larger is not sufficient

Front 74

significance level

Back 74

a point where if the p value is below it, then the null hypothesis is rejected, or if it is above, then it can not be rejected

Front 75

what is the notation for significance

Back 75

alpha

Front 76

what are the steps to hypothesis testing

Back 76

1. define the variable of interest and the population of interest. determine if the VI is quantitative or categorical.

2. determine the null hypothesis and alternative hypothesis

3. choose an appropriate hypothesis test for the null hypothesis and list the assumptions made by the hypothesis test.

4. find the p value

5. Conclusion write a sentnece in the context of the problem that answers the question of interest.

Front 77

null hypothesis

Back 77

a statement of no effect or no difference

Front 78

what is the notation for the null hypothesis

Back 78

H0

Front 79

what are some general rules for a null hypothesis

Back 79

always involves an equal sign

the population parameter, ux, will be the same as the hypothesized value, u0

Front 80

alternative hypothesis

Back 80

often associated with the researcheres claim or question. It makes a statement hypothesizing some change that will be expected to happen from a given treatment.

Front 81

what does a normal probability plot show

Back 81

the normality of a data set regardless of size

Front 82

what does not equal to impluy

Back 82

less than or greater than a value

Front 83

one tail test vs two tail test

Back 83

one tail test either looks @ one end or the other of a normal distribution

two tail test looks at both sides of a normal distribution, usually a given distance away from the mean.

Front 84

How do you find the p value for a two tailed test

Back 84

add together the probability of both tails

Front 85

what is an easy way to find a probability for a two tailed test

Back 85

find the probability of one tail and multiply by 2

Front 86

what is the type 1 hypothesis error

Back 86

occurs when there is evidence to reject the null hypothesis when the null hypothesis is actually true

Front 87

what is the type 2 hypothesis error

Back 87

when the null hypothesis is not rejected when it is actually false

Front 88

what is the probability of making a type 1 error if the significance level is given

Back 88

P(making error) = alpha

Front 89

what is the probability of making a type 1 error if the significane level is not given

Back 89

P(making error) = p value

Front 90

how do you determine the probability of making a type II error

Back 90

P(type II error) = beta and beta depends on the effect size

Front 91

effect size

Back 91

choosing a specific value for u that would make the null hypothesis and be large enough to make a difference

Front 92

power

Back 92

probability of correctly rejecting the null hypothesis

Front 93

how does one obtain a higher power value

Back 93

more sample size

Front 94

power is complements with...

Back 94

beta

Front 95

beta depends on

Back 95

a sample mean u that would cause one to accept the null hypothesis when it is not true

Front 96

equation for power

Back 96

1 - B

Front 97

point estimate

Back 97

using the sample mean as the best estimate of the population mean

Front 98

margin of error

Back 98

range of values above and below the sample mean that give potential values for the population parameter

Front 99

purpose of confidence interval

Back 99

it is used when trying to find out characteristics of a given population. In using confidence intervals, an assumption is made that the sample mean is close to the population mean. The confidence interval is a measure of the reliability of the estimate (usually the sample mean) and provides a range for which possible true values for the population parameter lie

Front 100

what is a requirement of using confidence intervals

Back 100

that the data be distributed normally

Front 101

what is the confidence interval surround

Back 101

the sample mean

Front 102

what is a 68% confidence interval

Back 102

the range is determined to be 1 SD above and below the sample mean. This is the range where 68% of the data is believed to fall

Front 103

what is the general formula for a confidence interval

Back 103

lower bound = sample mean - (# of sd away from the mean)(value of SD)

upper bound same as above but with a plus sign

Front 104

what does a 95% confidence interval imply

Back 104

that 95% of the population data is within 2 (*actually it is 1.96) sds above or below the sample mean

Front 105

how do you write a confidence interval

Back 105

(lower limit # & units, upper limit # & units)

Front 106

what is the critical value for a confidence interval and what does it depend on?

Back 106

critical value is z and it depends on the level of confidence

Front 107

what conditions must be met to use a confidence interval

Back 107

sample must be representative of the population
distribution of the sample means must be normal

observations must be independent

Front 108

what does the confidence interval states about many samples and their confidence intervals

Back 108

that from all the samples, 95% (or whatever conf value) will have a confidence interval that contains the true population mean

Front 109

when should a confidence interval be included

Back 109

whenever a hypothesis test is performed

Front 110

what is the relation between a two sided hypothesis test and the confidence interval at a given significance level

Back 110

If the hypothesized null hypothesis value is in the confidence interval, then there is no evidence to reject the null hypothesis at that significance level.

If the hypothesized value for the null hypothesis is not within the confidence interval, then there is evidence to reject the null hypothesis

Front 111

standard error of the distribution of sample means

Back 111

sample standard deviation / number of individuals in the sample

it is used when you do not know the standard deviation of the population

Front 112

t value equation

Back 112

mean - value / SEx

Front 113

t statistic represents

Back 113

the number of standard errosr from an observed value from the mean of the distribution

Front 114

as the sample size increases what happens to sx in terms of ox

Back 114

sx becomes a better estimate of ox

Front 115

what the point of degrees of freedom

Back 115

to account for the different sample means that will occur from different sample sizes

Front 116

degrees of freedmo

Back 116

n-1

Front 117

what is needed to calculate a p value in a t stat

Back 117

t stat and deg of freed

Front 118

a higher confidence means less

Back 118

precision

Front 119

what does the sahpe of a t distribution depend on

Back 119

the degrees of freedom

Front 120

can t stats be negative

Back 120

yes, the represent the # of standard errors below the mean

Front 121

where is the t value in a t value chart

Back 121

it makes up the bulk of it.

Front 122

If one were to look at a t stat of 2.492 what does the probability for that value mean?

Back 122

It is the probability of getting a t stat greater than the one you currently have

P(x>2.492)

Front 123

what are properties of a t distribution

Back 123

symmetric and unimodal

Front 124

how should one find the probaibliyt of a negative t stat

Back 124

find the positive one, they are they same

Front 125

what should one do when the t statistic can't be found on the table

Back 125

create an interval between the two closest values that surround your desired t stat

Front 126

when is it best to use a t test

Back 126

when the population standard deviation is NOT KNOWN

Front 127

what is the equation for the confidence interval on a t test

Back 127

mean +/- t value @ n-1 degrees of freedom *SEx

Front 128

what is a synonym for variable of interest

Back 128

response variable

Front 129

in a two sample t test how many populations are thewre

Back 129

2

Front 130

what is the purpose of a two sample t test

Back 130

it compares the means between two independent populations that have a response variable in common

Front 131

how should one write a null hypothesis for a 2 sample t test

Back 131

either than u1 = u2 or u1-u2 = 0

Front 132

how should one write the alternative hypothesis for a 2 sample t test

Back 132

using an inequality sign

ex: u1 > u2 or u1 - u2 > 0

Front 133

t are the conditions of a two sample t test

Back 133

the samples are representative of the population

the distribution of the differences betweetn the sample means is approx normal

the observations are independent of each other

Front 134

u vs x bar

Back 134

population parameter vs sample mean statistic

Front 135

what distribution does a two sample t test look at

Back 135

the distribution of the difference between the sample means of the two populations

Front 136

what does the standard error of the difference in the sample means represent

Back 136

the variance of the two populations added together, sqrted

Front 137

variance

Back 137

standard dev squared

Front 138

t stat for a 2 sample t tests

Back 138

observed - hypothethical / standard error

Front 139

how do you find the degrees of freedom for a two sample t test

Back 139

the smaller of the two sample sizes minus 1

n(smallest #) - 1

Front 140

how do you find the confidence interval for a two sample t test

Back 140

estimate between diff of population means +/- (t stat)(standard error)

Front 141

frequency table

Back 141

shows the frequency of individuals in a category

Front 142

contingency table

Back 142

a table that compares the frequency of the categorical variable of interests between 2 (or more) groups

Front 143

marginal distribution

Back 143

number of individuals in each category divided by the total

Front 144

conditional distribution

Back 144

same as a marginal distribution, only that it shows that of the group, how the distribution is broken up into the categories

ex:

38% of women felt overweight
34% of women felt just right

etc

Front 145

when the variable of interest is categorical, what is the random variable

Back 145

discrete

Front 146

expected value is synonymous with

Back 146

mean

Front 147

how to calculate the expected mean

Back 147

it is the categorical value times the probability of that value all summed together

Front 148

how to calculate the standard deviation

Back 148

the categorical value minus the mean squared times the probability summed together and square rooted

Front 149

what are the four conditions that must be met for a random varaible to have binomial distribution?

Back 149

1. each observation has 2 possible outcomes of which only 1 occurs
2. there is a fixed number of observations (n the sample size
3. the outcomes of the observations are independent
4. the probability of of an outcome occuring is the same in a single trial

Front 150

0! =?

Back 150

1

Front 151

what is a requirement of probability distributions

Back 151

that they will sum up to equal 1

Front 152

how to calculate the mean of a binomial random variable X

Back 152

ux = n*p

Front 153

what is the standard deviation of a binomial random variable

Back 153

ox = sqrt(n*p*(p-1)

Front 154

what is a geometric distribution show

Back 154

it is for a discrete random variable

it is the probability distribution of how many trials must happen before the first "success" occurs, or before the desired outcome occurs

Front 155

what does x represent in a geometric distribution

Back 155

the number of trials to the first "success"

Front 156

what is an equation to represent the probability of x in a geometric distribution

Back 156

P(x) = (1-p)^x-1 * p

Front 157

how is the expected value calculated in the geometric distribution

Back 157

ux = 1/p

Front 158

how is the standard deviation calculated for the geometric distribution

Back 158

ox = sqrt(I1-p)/p2)

Front 159

bernoulli trial

Back 159

is an experiment in which only two outcomes are possible

Front 160

binomial random variable

Back 160

the number of successes in a series of independent bernoulli trials

Front 161

when the sample size increases what happens to a bionomial distribution

Back 161

it begins to look more like a normal distribution

Front 162

what is a problem about computing the probability with a large sample size

Back 162

it is computationally intense

Front 163

what does the distribution of the sample proportions represent

Back 163

the distribution of lots of sample proportions of sample size n taken from a population with a true proportion p

Front 164

as the sampling size increases how does the sampling proportion and the population proportion relate to each other

Back 164

p hat = p

Front 165

what happens to the distribution of sampling proportions as the sample size increases

Back 165

it becomes smaller

Front 166

what is the sampling proportion distribution centered around

Back 166

p the true population proportion

Front 167

If we do not know p, the population proportion, what should one do

Back 167

use p hat in its place

Front 168

when using a sampling proportion distribution what should you check

Back 168

if np >10 and if n(1-p) >10

Front 169

how do you calculate the z statistic for a sampling proportion

Back 169

p hat - p all over standard deviation

Front 170

bernoulli trial

Back 170

situations where there are two outcomes: success or fail