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, etc) gave the following P/E ratios.

24, 16, 22, 14, 12, 13, 17, 22, 15, 19, 23, 13, 11, 18

Generally speaking, a low P/E ratio indicates a "value" or bargain stock. Financial publications indicated that the P/E ratio of the S&P 500 stock index has typically been 20.0. 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 = 4.1. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 20.0? Use α = 0.05.

(a) Enter the following. x =

s = (

b) Identify the claim, the null hypothesis, and the alternative hypothesis.

Claim: 20.0

Ho: 20.0

H1: 20.0

(c) Will you use a left-tailed, right-tailed, or two-tailed test? two-tailed left-tailed right-tailed

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

The Student's t, since we assume that x has a normal distribution with known σ.

The standard normal, since n is large with known σ.

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

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

The standard normal, since n is large with unknown σ. The Student's t, since n is large with unknown σ.

(e) Sketch the sampling distribution showing the area corresponding to the approximate P-value.

(f) Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

    At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant

.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

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

Fail to reject the null hypothesis, there is insufficient evidence that average P/E for large banks is less than the S&P 500 Index.

Fail to reject the null hypothesis, there is sufficient evidence that average P/E for large banks is less than the S&P 500 Index.    

  Reject the null hypothesis, there is insufficient evidence that average P/E for large banks is less than the S&P 500 Index

.Reject the null hypothesis, there is sufficient evidence that average P/E for large banks is less than the S&P 500 Index.

Answer 1

Solution:

Given: A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, etc) gave the following P/E ratios.

24, 16, 22, 14, 12, 13, 17, 22, 15, 19, 23, 13, 11, 18

We assume that x has a normal distribution and σ = 4.1.

We have to test if data indicate that the P/E ratio of all U.S. bank stocks is less than 20.0

Level of significance = α = 0.05

Part a) Enter the following.

We have to find sample mean and sample standard deviation.

thus we need to make following table:

x	x^2
24	576
16	256
22	484
14	196
12	144
13	169
17	289
22	484
15	225
19	361
23	529
13	169
11	121
18	324

Part b) Identify the claim, the null hypothesis, and the alternative hypothesis

Claim:

Part c) Will you use a left-tailed, right-tailed, or two-tailed test?

Left-tailed . since H1 is < type.

Part d) 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.

Part e) Sketch the sampling distribution showing the area corresponding to the approximate P-value.

We need to find z test statistic and P-value.

P-value = P( Z < -2.67)

From z table , for z = -2.6 and 0.07 , area is: 0.0038

That is: P( Z < - 2.67) = 0.0038

This P-value = 0.0038

Part f) Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

Since P-value = 0.0038 < 0.05 level of significance , we reject H0. Thus :

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant

Part g) State your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence that average P/E for large banks is less than the S&P 500 Index.