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23 Cards in this Set
 Front
 Back
Uniform probability distribution

1 / (ba) where a<x<b
to get the probability within the interval, multiply the segment of the interval by 1/(ba) 

standard deviations on the normal distribution curve

1 SD covers 68.3%
2 SD covers 95.4% 3 SD covers 99.7% 

Variance

(xi  ẍ)^2 / N1 (sample)
(xi  U)^2 / N (population) The sigma notation indicates that if there are a group of values, to take the total variance find the sum of deviations squared and divide 

Xi

number for one element in the set


ẍ

sample mean


Point Estimation

for certain problems, the point estimate of a set will be the sample mean ẍ
it can also be the proportion of values calculated with regard to a specific question, such as the proportion of managers who attended a certain class 

Standard deviation of sampling distributions (ẍ)
a.k.a. standard error of the mean 
For finite populations:
(√Nn / √N1) * (SD / √n) SD=standard deviation of the population, n=sample size, N=population size For infinite populations: SD / √n 

Standard deviation of ṕ

Finite Population:
(√Nn / √N1) * (√p(1p) / √n) Infinite: (√p(1p) / √n 

Interval estimate of a population mean where SD is known

ẍ ± Z(a/2)* SD / √n
where Z(a/2) = 1.96 at 95% confidence, 1.645 at 90%, 2.576 at 99% You multiply the equation and then BOTH add and subtract the value with ẍ to give the interval value 

Interval estimate of a population mean where SD is unknown

ẍ ± t(a/2)* s/√n
a/2 is the area in the upper tail, s is the sample SD. for the ttables, use n1 degrees of freedom (n is given) 

How to get the sample size for the interval estimate of a population mean

n = (z(a/2)^2*(SD^2)) / E^2
z(a/2) is the z value (z(.025) = 1.96, etc) E will be given 

how to get the interval estimate for a population proportion

p ± z(a/2) * (√p(1p)) / √n
p is the proportion of the total population to the smaller pop. given based on the question, i.e. x / n with p ± x (whatever the answer is) the answer is the margin of error and subtracting and adding it to p gives the interval 

how to find the sample size for an interval estimate of a population proportion

n = (z(a/2)^2 * p'(1p')) / E^2
p' is the planning value, which can be found either using a previous sample proportion, or likely using .5 

basics of Null and alternative hypotheses, denoted by Ho and Ha

≥ or ≤ ALWAYS appear in the null hypothesis
If we reject Ho and it is true, it is a Type I error If we accept Ho and it is false, it is a Type II error Counterintuitively, upper tailed tests ≤ and lower tailed tests feature ≥ in Ho two tailed tests involve two outcomes: where the correct outcome/value is acheived, or it is not 

Finding the test statistic for hypothesis tests about a pop. mean when SD is known

z = (ẍ  u) / (SD / √n)
u = population mean, the value which you are testing against 

one tailed hypothesis tests using the pvalue approach

use the normal probability table and plug in the test statistic value. we reject Ho if the pvalue is less than α


one tailed hypothesis tests using the critical value approach

reject Ho if z (test statistic) is less than or equal to z(a)
this means we find the value on the left side of the table which will give the α given in the problem (.01. .05, etc) and determine whether the calculated test statistic is less than or equal to that value 

pvalue hypothesis tests for two tailed tests

same equation as with one tailed tests. take that value of z and multiply by 1, so that if your value is 1.53 you have 1.53. look up the value and multiply by 2 for the pvalue. if the p value is less than or equal to a, reject Ho.


critical value hypothesis test for two tailed tests

we reject Ho if z is less than or equal to z(a/2) or greater than z(a/2). that is, based on the value of a find the corresponding value on the left side of the table, and compare to the calculated statistic for z. if a =.05, then z(a/2) = z.025 = 1.96


hypothesis testing with confidence intervals

ẍ ± z(a/2)* SD/√n
the answer ẍ ± ____ gives you the confidence interval. if the population mean falls within the interval, do not reject Ho. 

hypothesis tests with SD unknown, one tailed

test statistic equation stays the same. take that value and look it up on the t table with n1 degrees of freedom to give you the pvalue. if the p value is less than a, reject Ho.


hypothesis tests with SD unknown, two tailed

equation stays the same, plug the value into the ttables and determine the area in upper tail where t falls. if that value is less than a, reject Ho.


test statistic for hypothesis tests about a population proportion

z = ṕ  po / (√po(1po) / √n)
ṕ = the population proportion (sample/total, i.e n) po = the value being tested against in Ho/Ha 