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 = 3.9. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 18? Use α = 0.10.

(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?

(1) H₀: μ = 18; H₁: μ < 18; left-tailed

(2) H₀: μ ≠ 18; H₁: μ = 18; two-tailed    

(3) H₀: μ = 18; H₁: μ ≠ 18; two-tailed

(4) H₀: μ = 18; H₁: μ > 18; right-tailed

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

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 = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) State your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the P/E ratio of all large U.S. bank stocks is less than 18.

There is insufficient evidence at the 0.10 level to conclude that the P/E ratio of all large U.S. bank stocks is less than 18.   

Answer 1

Given that, sample size (n) = 14

sample mean = 17.1

population standard deviation = 3.9

a) Level of significance = α = 0.10

The null and alternative hypotheses are,

H0 : μ = 18; H1 : μ < 18; left-tailed

b) The sampling distribution of the sample mean is approximately normal because, population standard deviations is known.

Test statistic is,

=> Test statistic = Z = -0.86

c) p-value = P(Z < -0.86) = 0.1949

=> p-value = 0.1949

d) At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e) Conclusion : There is insufficient evidence at the 0.10 level to conclude that the P/E ratio of all large U.S. bank stocks is less than 18.