EXAMPLE 20 Test using the -value method: Right-tailed test

milkprice

The U.S. Bureau of Labor Statistics reports that the mean price for a gallon of milk in July 2014 was $3.65. Gallons of milk were recently bought in a random sample of different cities, with the prices shown in the accompanying table. Test, using level of significance , whether the population mean price for a gallon of milk is greater than $3.65.

Solution

We first check whether the conditions for performing the test are met. Because our sample size is small, we must check for normality. The normal probability plot in Figure 22 shows acceptable normality, allowing us to proceed with the test.

image
Figure 9.27: FIGURE 22 Normal probability plot of milk prices.

  • Step 1 State the hypotheses and the rejection rule.

    The key words “is greater than” means that we have a right-tailed test. Answering the question “Greater than what?” gives us .

    where represents the population mean price of milk. We will reject if the .

  • Step 2 Calculate .

    We use the instructions from the Step-by-Step Technology Guide on page 537. Figure 23 shows the TI-83/84 results from the test for .

    image
    Figure 9.28: FIGURE 23 TI-83/84 results for right-tailed test.

    For a more accurate calculation of the -value, we retain 9 decimal places for the value of .

    Using the statistics from Figure 23, we have the test statistic

  • Step 3 Find the -value.

    From Figures 23 and 24, we have

    531

    image
    Figure 9.29: FIGURE 24 The -value for a right-tailed test.

  • Step 4 State the conclusion and the interpretation.

    The is less than the level of significance ; therefore, reject . There is evidence, at level of significance , that the population mean price of milk is greater than $3.65.

NOW YOU CAN DO

Exercises 15–20.