The price to earnings ratio (P/E) is an important tool in financial work. A rand

Question

The price to earnings ratio (P/E) is an important tool in financial work. A random sample of 14 targe 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 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 had a population mean of 20. 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 the population standard deviation is 4.9. Does this data indicate that the P/E ratio of all U.S. bank stocks is less than 20? use alpha = 0.01.

Explanation / Answer

Here, we have to use the one sample z test for the population mean.

The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: The average P/E ratio of all U.S. bank stocks is 20.

Alternative hypothesis: Ha: The average P/E ratio of all U.S. bank stocks is less than 20.

H0: µ = 20 versus Ha: µ < 20

We are given, level of significance = alpha = 0.01

Sample mean = 17.1

Population standard deviation = 4.9

The test statistic formula is given as below:

Z = (xbar – mean) / [ / sqrt(n)]

Z = (17.1 – 20) / [4.9 / sqrt(14)] = -2.2145

Z Test of Hypothesis for the Mean

Data

Null Hypothesis                       m=

20

Level of Significance

0.01

Population Standard Deviation

4.9

Sample Size

14

Sample Mean

17.1

Intermediate Calculations

Standard Error of the Mean

1.3096

Z Test Statistic

-2.2145

Lower-Tail Test

Lower Critical Value

-2.3263

p-Value

0.0134

Do not reject the null hypothesis

Here, we get the p-value as 0.0134 which is greater than the given level of significance or alpha value 0.01, so we do not reject the null hypothesis that the average P/E ratio of all U.S. bank stocks is 20.

Z Test of Hypothesis for the Mean

Data

Null Hypothesis                       m=

20

Level of Significance

0.01

Population Standard Deviation

4.9

Sample Size

14

Sample Mean

17.1

Intermediate Calculations

Standard Error of the Mean

1.3096

Z Test Statistic

-2.2145

Lower-Tail Test

Lower Critical Value

-2.3263

p-Value

0.0134

Do not reject the null hypothesis

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