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23 Cards in this Set
- Front
- Back
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Uniform probability distribution
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1 / (b-a) where a<x<b
to get the probability within the interval, multiply the segment of the interval by 1/(b-a) |
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standard deviations on the normal distribution curve
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1 SD covers 68.3%
2 SD covers 95.4% 3 SD covers 99.7% |
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Variance
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(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 |
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Xi
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number for one element in the set
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ẍ
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sample mean
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Point Estimation
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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 |
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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 |
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Standard deviation of ṕ
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Finite Population:
(√N-n / √N-1) * (√p(1-p) / √n) Infinite: (√p(1-p) / √n |
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Interval estimate of a population mean where SD is known
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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 |
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Interval estimate of a population mean where SD is unknown
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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) |
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How to get the sample size for the interval estimate of a population mean
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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 |
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how to get the interval estimate for a population proportion
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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 |
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how to find the sample size for an interval estimate of a population proportion
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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 |
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basics of Null and alternative hypotheses, denoted by Ho and Ha
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≥ 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 |
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Finding the test statistic for hypothesis tests about a pop. mean when SD is known
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z = (ẍ - u) / (SD / √n)
u = population mean, the value which you are testing against |
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one tailed hypothesis tests using the p-value approach
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use the normal probability table and plug in the test statistic value. we reject Ho if the p-value is less than α
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one tailed hypothesis tests using the critical value approach
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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 |
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p-value hypothesis tests for two tailed tests
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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.
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critical value hypothesis test for two tailed tests
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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
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hypothesis testing with confidence intervals
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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. |
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hypothesis tests with SD unknown, one tailed
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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.
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hypothesis tests with SD unknown, two tailed
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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.
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test statistic for hypothesis tests about a population proportion
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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 |
