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 = 18. 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 = 4.4. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 18? Use α = 0.01.

(a) What is the level of significance?


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ = 18; H₁: μ < 18; left-tailedH₀: μ = 18; H₁: μ > 18; right-tailed    H₀: μ = 18; H₁: μ ≠ 18; two-tailedH₀: μ ≠ 18; H₁: μ = 18; two-tailed

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The standard normal, since we assume that x has a normal distribution with known σ.The Student's t, since n is large with unknown σ.    The standard normal, since we assume that x has a normal distribution with unknown σ.The Student's t, since we assume that x has a normal distribution with known σ.

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


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


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Answer 1

The statistic software output for this problem is:

One sample Z summary hypothesis test: ean of population u: M Ho u 18 HA u 18 Standard deviation = 4.4 Hypothesis test results: Mean n Sample Mean Std. Err. P-value Z-Stat 14 17.1 1.1759495 -0.76533901 0.222

(a)  the level of significance = 0.01

H₀: μ = 18; H₁: μ < 18

(b)

The standard normal, since we assume that x has a normal distribution with known σ

z = -0.77

(c) P-value = 0.22220

Graph IV

(d)

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.