## The hypothesis test for analysis of variance for populations:

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### Next section: to Inferential statistics (testing hypotheses)

Because both samples are large (__>__ 30), we can use the Z test statistic as opposed to t. Note that statistical computing packages use t throughout. Before implementing the formula, we first check whether the assumption of equality of population variances is reasonable. The guideline suggests investigating the ratio of the sample variances, s_{1}^{2}/s_{2}^{2}. Suppose we call the men group 1 and the women group 2. Again, this is arbitrary; it only needs to be noted when interpreting the results. The ratio of the sample variances is 17.5^{2}/20.1^{2} = 0.76, which falls between 0.5 and 2 suggesting that the assumption of equality of population variances is reasonable. The appropriate test statistic is

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### The test statistic for testing H0: μ1 = μ2 = ... = μk is:

Because we are assuming equal variances between groups, we pool the information on variability (sample variances) to generate an estimate of the variability in the population. Note: Because Sp is a weighted average of the standard deviations in the sample, Sp will always be in between s_{1} and s_{2}.)

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We now substitute the sample data into the formula for the test statistic identified in Step 2. We first compute the overall proportion of successes:

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## How to Set Up a Hypothesis Test: Null versus Alternative

Recall that when we fail to reject H_{0} in a test of hypothesis that either the null hypothesis is true (here the mean expenditures in 2005 are the same as those in 2002 and equal to $3,302) or we committed a Type II error (i.e., we failed to reject H_{0} when in fact it is false). In summarizing this test, we conclude that we do not have sufficient evidence to reject H_{0}. We do not conclude that H_{0} is true, because there may be a moderate to high probability that we committed a Type II error. It is possible that the sample size is not large enough to detect a difference in mean expenditures.

## I’m stuck on how to value the null or alternative hypotheses

**Born in Mustamaki,** Finland, Hoeffding began his university education studying economics but quickly switched to mathematics eventually earning a Ph.D. degree from Berlin University in 1940 with a dissertation on nonparametric measures of association and correlation. He emigrated to the USA in 1946 settling in Chapel Hill, North Carolina. Hoeffding made significant contributions to sequential analysis, statistical decision theory and central limit theorems. He died on 28 February 1991 in Chapel Hill.

## What descriptive and inferrential statistics to use

We do not reject H_{0} because -1.26 > -1.645. We do not have statistically significant evidence at α=0.05 to show that the mean expenditures on health care and prescription drugs are lower in 2005 than the mean of $3,302 reported in 2002.

## hypothesis stating that the mean is ..

A distribution free method for testing for the independence of two random variables X and Y, that is able to detect a broader class of alternatives to independence than is possibly by using sample correlation coefficients. [NSM Chapter 8.]