# Hypothesis testing for means and proportions

A representative sample of salaries for women in a particular job classification yields the following results: This is a one-tailed hypothesis test to the left because we are seeing if there is strong evidence that the mean is less than some value. Most investigators are very comfortable with this and are confident when rejecting H0 that the research hypothesis is true as it is the more likely scenario when we reject H0. When we run a test of hypothesis and decide not to reject H0 e. When we do not reject H0, it may be very likely that we are committing a Type II error i.

Therefore, when tests are run and the null hypothesis is not rejected we often make a weak concluding statement allowing for the possibility that we might be committing a Type II error. If we do not reject H0, we conclude that we do not have significant evidence to show that H1 is true.

We do not conclude that H0 is true. The most common reason for a Type II error is a small sample size. Tests with One Sample, Continuous Outcome Hypothesis testing applications with a continuous outcome variable in a single population are performed according to the five-step procedure outlined above.

A key component is setting up the null and research hypotheses. The known value is generally derived from another study or report, for example a study in a similar, but not identical, population or a study performed some years ago.

The latter is called a historical control. It is important in setting up the hypotheses in a one sample test that the mean specified in the null hypothesis is a fair and reasonable comparator.

This will be discussed in the examples that follow. In one sample tests for a continuous outcome, we set up our hypotheses against an appropriate comparator. We select a sample and compute descriptive statistics on the sample data - including the sample size nthe sample mean and the sample standard deviation s.

We then determine the appropriate test statistic Step 2 for the hypothesis test. The formulas for test statistics depend on the sample size and are given below. Test Statistics for Testing H0: Data are provided for the US population as a whole and for specific ages, sexes and races.

An investigator hypothesizes that in expenditures have decreased primarily due to the availability of generic drugs.

## Comparing two proportions

To test the hypothesis, a sample of Americans are selected and their expenditures on health care and prescription drugs in are measured. The sample data are summarized as follows: Is there statistical evidence of a reduction in expenditures on health care and prescription drugs in ? We will run the test using the five-step approach. Set up hypotheses and determine level of significance H0: Select the appropriate test statistic. Set up decision rule.

• Hypothesis Testing for Two Means and Two Proportions (Worksheet) - Statistics LibreTexts
• Hypothesis Testing For A Population Proportion

Compute the test statistic. We now substitute the sample data into the formula for the test statistic identified in Step 2. We do not reject H0 because In summarizing this test, we conclude that we do not have sufficient evidence to reject H0.

We do not conclude that H0 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.

The NCHS reported that the mean total cholesterol level in for all adults was Total cholesterol levels in participants who attended the seventh examination of the Offspring in the Framingham Heart Study are summarized as follows: Is there statistical evidence of a difference in mean cholesterol levels in the Framingham Offspring?

Here we want to assess whether the sample mean of We reject H0 because Because we reject H0, we also approximate a p-value. Statistical Significance versus Clinical Practical Significance This example raises an important concept of statistical versus clinical or practical significance. However, the sample mean in the Framingham Offspring study is The reason that the data are so highly statistically significant is due to the very large sample size.

It is always important to assess both statistical and clinical significance of data.CHAPTER 9. HYPOTHESIS TESTING: SINGLE MEAN AND SINGLE PROPORTION 1.