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 = 4.5. 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 hypothesis
H0
and the alternate hypothesis
H1
.
H0
: μ ≥ 18
H1
: μ < 18
What kind of test is this?
left-tailed test
(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.
Compute the sample test statistic based on your choice of the distribution. (Round your answer to two decimal places.)
(c) Find 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 the previous parts, state your decision. 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 fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) State your conclusion.
There is insufficient evidence at the 0.01 level to conclude that the P/E ratio of all large U.S. bank stocks is less than 18.