Comparing Two Means and Two Standard Deviations Module 23

Comparing Two Means We use the same basic principles for comparing two population means as those used for examining one population mean. If the standard deviations  1 and  2 for each of the two populations are known, the two-sample z-statistic is then But it is very rare that both population standard deviations are known. We will examine the situation in which they are not known.

Comparing Two Means When we are interested in comparing two population means and we are estimating the population standard deviations  1 and  2 with s 1 and s 2, the two-sample t- statistic is then

Comparing Two Means The null hypothesis can be any of the following: The alternative hypothesis can be any of the following (depending on the question being asked): The other steps are the same as those used for the tests we have looked at previously.

Comparing Two Means Example: Tomatoes –“There has been some discussion among amateur gardeners about the virtues of black plastic versus newspapers as weed inhibitors for growing tomatoes. To compare the two, several rows of tomatoes are planted. Black plastic is used around nine randomly selected plants and newspaper around the remaining ten. All plants start at virtually the same height and receive the same care. The response of interest is the height in feet after a month’s growth.” (from Milton, McTeer, and Corbet, Introduction to Statistics, 1997). –Perform a test to see if there is any difference between the average heights with significance level 0.10.

Example: Tomatoes –Information given: Sample sizes: n 1 = 9, n 2 = 10. Comparing Two Means

Example: Tomatoes –1. State the null hypothesis: –2. State the alternative hypothesis: –3. State the level of significance Comparing Two Means

Example: Tomatoes –4. Calculate the test statistic. –5. Find the P-value. Comparing Two Means

Example: Tomatoes –6. Do we reject or fail to reject H 0 based on the P- value? –7. State the conclusion. Comparing Two Means

The confidence interval for the difference of two population means (  1 -  2 ) is Where t* comes from Table D and corresponds to the confidence level desired and df = smaller of n 1 -1 and n 2 -1.

Comparing Two Means Example: Commercials –“There is some concern that TV commercial breaks are becoming longer. The observations on the following slide are obtained on the length in minutes of commercial breaks for the 1984 viewing season and the current season.” (from Milton, McTeer, and Corbet, Introduction to Statistics, 1997) –Find a 95% confidence interval for the difference between the true averages of the two seasons.

Example: Commercials –Information given: Sample sizes: n 1 = 16, n 2 = 16. Comparing Two Means

Example: Commercials t* is found in table D. We first go to the 95% confidence level at the bottom. Then we go up to 15 df. Thus, t* = Comparing Two Means

Comparing Two Standard Deviations To compare two population standard deviations, we use the ratio of the sample variances. We call this a test for equality of spread. The null and alternative hypotheses are

Comparing Two Standard Deviations The test statistic is the F statistic. where s 1 and s 2 come from samples of size n 1 and n 2. s 1 2 is chosen to be the larger of the two sample variances.