Chapter 9: Hypothesis Testing with Two Samples
Introduction to Chapter 9: Hypothesis Testing with Two Samples
Studies often compare two groups. For example, researchers are interested in the effect aspirin has in preventing heart attacks. Over the last few years, newspapers and magazines have reported various aspirin studies involving two groups. Typically, one group is given aspirin and the other group is given a placebo. Then, the heart attack rate is studied over several years.
There are other situations that deal with the comparison of two groups. For example, studies compare various diet and exercise programs. Politicians compare the proportion of individuals from different income brackets who might vote for them. Students are interested in whether SAT or GRE preparatory courses really help raise their scores.
You have learned to conduct hypothesis tests on single means and single proportions. You will expand upon that in this chapter. You will compare two means or two proportions to each other. The general procedure is still the same, just expanded.
To compare two means or two proportions, you work with two groups. The groups are classified either as independent or matched pairs. Independent groups consist of two samples that are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Matched pairs consist of two samples that are dependent. The parameter tested using matched pairs is the population mean. The parameters tested using independent groups are either population means or population proportions.
This chapter deals with the following hypothesis tests:
- Test of two population means.
- Test of two population proportions.
- Test of the two population proportions by testing one population mean of differences.
"Traditional Statistics" versus Technology Note
The traditional method of calculating degrees of freedom, the test statistic, and p-value for hypothesis testing of two samples using by-hand methods with formulas and statistical tables is covered in the main sections of the text.
An added section titled Using Technology is provided at the end of the chapter that provide instructions for calculating values to do hypothesis testing of two samples by using the TI-83+ and TI-84 calculators and other technology tools.
**When using a TI-83+ or TI-84 calculator, we do not need to separate two population means, independent groups, or population variances unknown into large and small sample sizes. However, most statistical computer software has the ability to differentiate these tests. **
Hypothesis Testing with Two Samples - Solution Sheet
To complete the homework in this chapter, use the following solution sheet.
- H0: _______
- Ha: _______
- In words, clearly state what your random variable \({\overline{X}}_{1}-{\overline{X}}_{2}\), \({{P}^{\prime }}_{1}-{{P}^{\prime }}_{2}\) or \({\overline{X}}_{d}\) represents.
- State the distribution to use for the test.
- What is the test statistic?
- What is the p-value? In one to two complete sentences, explain what the p-value means for this problem.
- Use the previous information to sketch a picture of this situation on Figure a. Clearly label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
Figure a. Normal curve to sketch a picture of the p-value - Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
- Alpha: _______
- Decision: _______
- Reason for decision: _______
- Conclusion: _______
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