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23 Cards in this Set

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Uniform probability distribution
1 / (b-a) where a<x<b
to get the probability within the interval, multiply the segment of the interval by 1/(b-a)
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 / N-1 (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
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:
(√N-n / √N-1) * (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:
(√N-n / √N-1) * (√p(1-p) / √n)

Infinite:
(√p(1-p) / √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 t-tables, use n-1 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(1-p)) / √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'(1-p')) / 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 p-value approach
use the normal probability table and plug in the test statistic value. we reject Ho if the p-value 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
p-value 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 p-value. 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 n-1 degrees of freedom to give you the p-value. 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 t-tables 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(1-po) / √n)

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