91Ó°ÊÓ

Please provide the following information. (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? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (c) Find (or estimate) the \(P\) -value. 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 \(\alpha\) ? (e) State your conclusion in the context of the application. The price to earnings ratio \((\mathrm{P} / \mathrm{E})\) 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 \(\mathrm{P} / \mathrm{E}\) ratios (Reference: Forbes). \(\begin{array}{lllllll}24 & 16 & 22 & 14 & 12 & 13 & 17 \\\ 22 & 15 & 19 & 23 & 13 & 11 & 18\end{array}\) The sample mean is \(\bar{x} \approx 17.1\). Generally speaking, a low \(\mathrm{P} / \mathrm{E}\) ratio indicates a "value" or bargain stock. A recent copy of The Wall Street Journal indicated that the \(\mathrm{P} / \mathrm{E}\) ratio of the entire \(\mathrm{S\&P} 500\) stock index is \(\mu=19 .\) Let \(x\) be a random variable representing the \(\mathrm{P} / \mathrm{E}\) ratio of all large U.S. bank stocks. We assume that \(x\) has a normal distribution and \(\sigma=4.5\). Do these data indicate that the \(\mathrm{P} / \mathrm{E}\) ratio of all U.S. bank stocks is less than 19 ? Use \(\alpha=0.05\).

Step by step solution

01

Define the Hypotheses and Determine the Test Type

The level of significance is given as \( \alpha = 0.05 \). We need to test if the \( \mathrm{P} / \mathrm{E} \) ratio of all U.S. bank stocks is less than 19. Therefore, the null and alternate hypotheses are:\[ H_0: \mu = 19 \]\[ H_a: \mu < 19 \]Since we are checking if the \( \mathrm{P} / \mathrm{E} \) ratio is less than 19, this is a left-tailed test.

02

Determine the Sampling Distribution and Calculate the Test Statistic

The population standard deviation \( \sigma \) is known (\( \sigma = 4.5 \)), so we will use the normal distribution to determine the test statistic. The test statistic \( z \) is calculated as follows:\[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]Here, \( \bar{x} = 17.1 \), \( \mu = 19 \), \( \sigma = 4.5 \), and \( n = 14 \). Substituting these values in, we get:\[ z = \frac{17.1 - 19}{4.5 / \sqrt{14}} = \frac{-1.9}{1.202} \approx -1.58 \]

03

Estimate the p-Value and Sketch the Distribution

To find the \( p \)-value, we use the standard normal distribution table for \( z = -1.58 \). The \( p \)-value is the area to the left of \( z = -1.58 \). By consulting a \( z \)-table, we find that the \( p \)-value \( \approx 0.0571 \). Sketch a normal curve, locate \( z = -1.58 \) on the horizontal axis, and shade the area to the left of this point.

04

Make a Decision about the Null Hypothesis

Compare the \( p \)-value to the significance level \( \alpha = 0.05 \). Since \( 0.0571 > 0.05 \), we fail to reject the null hypothesis. The data is not statistically significant at the \( \alpha = 0.05 \) level.

05

Conclusion in the Context of the Application

Based on the analysis, there is insufficient evidence to conclude that the \( \mathrm{P} / \mathrm{E} \) ratio of all U.S. bank stocks is significantly less than 19 at the \( 5\% \) level of significance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significance Level

In hypothesis testing, the significance level, often denoted as \( \alpha \), is a threshold that helps us decide whether to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true, also known as the risk of Type I error. In simple terms, it defines how much evidence is needed to consider the result statistically significant.

Most common significance levels used are 0.05, 0.01, and 0.10. Here, \( \alpha = 0.05 \). This means we accept a 5% risk that we might make a Type I error. In our exercise, this threshold helps us decide if the observed data supports that the P/E ratio is, in fact, different from 19 for U.S. bank stocks.

• Type I error: Incorrectly rejecting the true null hypothesis.
• Type II error: Failing to reject a false null hypothesis.

Choosing a proper significance level is crucial as it influences the potential errors and confidence in the results.

Normal Distribution

The normal distribution is a fundamental concept in statistics, underlying many statistical tests, including hypothesis testing. It is a continuous probability distribution characterized by its bell shape and symmetric property about the mean. The mean, median, and mode of a normal distribution all are equal, and about 68% of the data lies within one standard deviation of the mean.

In our example, the sample’s standard deviation \( \sigma = 4.5 \) is known, which enables us to use the normal distribution for our analysis. This is possible because the normal distribution is suitable for large sample sizes or when the population standard deviation is known, providing reliable results for our calculations.

• The curve is defined as:
  \[ f(x|\mu,\sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{- \frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 } \]
• 68% of values fall within \( \mu \pm \sigma \), 95% within \( \mu \pm 2\sigma \), and 99.7% within \( \mu \pm 3\sigma \).

This outlines why a known standard deviation and a normal distribution foundation enable us to calculate our test statistic effectively, driving the decision in hypothesis testing.

Null and Alternative Hypotheses

The null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \) are two opposing statements about a population parameter that we test using sample data. Establishing these hypotheses is the starting point in hypothesis testing.

The null hypothesis \( H_0 \) is a statement suggesting there is no effect or difference, serving as a default or baseline. In our example, \( H_0: \mu = 19 \), implying the P/E ratio of U.S. bank stocks is equal to 19. Meanwhile, the alternative hypothesis \( H_a \) proposes there is an effect or difference, represented by \( H_a: \mu < 19 \), indicating the P/E ratio is less than 19.

• \( H_0 \): Assumes no change - baseline or status quo.
• \( H_a \): Indicates a potential change - what the researcher aims to prove.

The type of test (left, right, two-tailed) depends on \( H_a \), guiding where to look for the evidence against \( H_0 \). Here, being a left-tailed test signifies searching for evidence in the direction where the P/E ratio is significantly less than 19.