Question

The price to earnings ratio (P/E) is an important tool in financial work. A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, and others) gave the following P/E ratios.†

24

16

22

14

12

13

17

22

15

19

23

13

11

18

The sample mean is

x ≈ 17.1.

Generally speaking, a low P/E ratio indicates a "value" or bargain stock. Suppose a recent copy of a magazine indicated that the P/E ratio of a certain stock index is μ = 19. Let x be a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that x has a normal distribution and σ = 5.0. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 19? Use α = 0.05.

A) Compute the z value of the sample test statistic. (Round your answer to two decimal places.)


B) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Answer 1

The provided sample mean is 17.1 and the known population standard deviation is σ=5, and the sample size is n = 14

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=19

Ha: μ<19

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is z_c = -1.64

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = -1.422 >z_c = -1.64, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.0775, and since p = 0.0775 > 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is less than 19, at the 0.05 significance level.

Hence the data indicate that the P/E ratio of all U.S. bank stocks is not less than 19