91Ó°ÊÓ

Finance: \(\mathrm{P} / \mathrm{E}\) Ratio The price of a share of stock divided by a company's estimated future earnings per share is called the P/E ratio. High P/E ratios usually indicate "growth" stocks, or maybe stocks that are simply overpriced. Low \(\mathrm{P} / \mathrm{E}\) ratios indicate "value" stocks or bargain stocks. A random sample of 51 of the largest companies in the United States gave the following P/E ratios (Reference: Forbes). $$ \begin{array}{rrrrrrrrrrrr} 11 & 35 & 19 & 13 & 15 & 21 & 40 & 18 & 60 & 72 & 9 & 20 \\ 29 & 53 & 16 & 26 & 21 & 14 & 21 & 27 & 10 & 12 & 47 & 14 \\ 33 & 14 & 18 & 17 & 20 & 19 & 13 & 25 & 23 & 27 & 5 & 16 \\ 8 & 49 & 44 & 20 & 27 & 8 & 19 & 12 & 31 & 67 & 51 & 26 \\ 19 & 18 & 32 & & & & & & & & & \end{array} $$ (a) Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x} \approx 25.2\) and \(s \approx 15.5\). (b) Find a \(90 \%\) confidence interval for the \(\mathrm{P} / \mathrm{E}\) population mean \(\mu\) of all large U.S. companies. (c) Find a \(99 \%\) confidence interval for the \(\mathrm{P} / \mathrm{E}\) population mean \(\mu\) of all large U.S. companies. (d) Interpretation Bank One (now merged with J.P. Morgan) had a P/E of 12 , AT\&T Wireless had a \(\mathrm{P} / \mathrm{E}\) of 72 , and Disney had a \(\mathrm{P} / \mathrm{E}\) of 24 . Examine the confidence intervals in parts (b) and (c). How would you describe these stocks at the time the sample was taken? (e) Check Requirements In previous problems, we assumed the \(x\) distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: See the central limit theorem in Section \(6.5\).

Step by step solution

01

Calculate the Mean

To calculate the sample mean \( \bar{x} \) of the given P/E ratios, add all the ratios and divide by the number of ratios. There are 51 data points, so \( \bar{x} = \frac{11 + 35 + 19 + ... + 32}{51} \approx 25.2 \).

02

Calculate the Sample Standard Deviation

The sample standard deviation \( s \) for the given data is calculated using \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \) where \( x_i \) represents each data point and \( n = 51 \). After computation, \( s \approx 15.5 \).

03

Find 90% Confidence Interval

For a 90% confidence interval, use the formula \( \bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}} \). With \( n = 51 \), \( t_{0.05} \approx 1.676 \) from the t-distribution table. Substitute to find the interval: \( 25.2 \pm 1.676 \cdot \frac{15.5}{\sqrt{51}} \approx (21.6, 28.8) \).

04

Find 99% Confidence Interval

For a 99% confidence interval, use \( \bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}} \). With \( n = 51 \), \( t_{0.005} \approx 2.678 \). The interval is \( 25.2 \pm 2.678 \cdot \frac{15.5}{\sqrt{51}} \approx (20.2, 30.2) \).

05

Interpret Confidence Intervals

Bank One has a P/E ratio at the lower bounds of both confidence intervals, indicating it could be a value stock. AT&T Wireless has a P/E significantly higher than the interval, suggesting it may be overvalued. Disney's P/E ratio is within both intervals, indicating it is in line with typical large U.S. companies.

06

Check Requirements with Central Limit Theorem

Due to the sample size (51) being greater than 30, the central limit theorem allows us to assume the sampling distribution of the mean is approximately normal, even without the original population distribution being normal.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values that estimates a population parameter, such as the mean, with a certain level of confidence. In the given problem, the task is to find confidence intervals for the P/E ratios of large U.S. companies. The 90% and 99% confidence intervals are computed using the sample mean, sample standard deviation, and the appropriate t-value from the t-distribution table. The intervals provide a range in which we are confident the true population mean lies, based on our sample data. The construction of a confidence interval involves:

• Calculating the sample mean (\(\bar{x}\))
• Using a sample standard deviation (\(s\))
• Determining the appropriate t-value for the confidence level

The formula is: \[\bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}}\]where \(t_{\frac{\alpha}{2}}\) is the t-value and \(n\) is the number of observations.Confidence levels like 90% or 99% reflect how sure we are about the interval including the true mean. A 90% confidence interval suggests that 90 out of 100 similar samples would contain the population mean. This concept helps in decision-making processes, like assessing stock valuation in this exercise.

Central Limit Theorem

The Central Limit Theorem (CLT) is fundamental in statistics because it allows us to make inferences about population parameters based on sample statistics, even when we do not know the population's distribution. According to the CLT, the distribution of the sample mean will tend to be normal or approximately normal if the sample size is sufficiently large, typically \(n > 30\). This holds true regardless of the initial distribution of data. In this exercise, the sample size is 51. Since it is greater than 30, the CLT assures us that the sampling distribution of the sample mean will be approximately normal. This is pivotal because:

• It justifies the use of t-distribution to calculate confidence intervals, even if the original population is not normal.
• Ensures that conclusions drawn about the population mean from the sample mean are reliable.

This theorem simplifies the analysis, allowing researchers and analysts to proceed with statistical methods that assume normality.

Sample Standard Deviation

Sample standard deviation is a measure of the amount of variation or dispersion in a set of values. It is denoted by \(s\) and is computed by taking the square root of the variance. Variance represents the average of squared differences from the mean. In this exercise, the sample standard deviation helps assess how much the P/E ratios deviate from the sample mean of 25.2. To find the sample standard deviation:

• Calculate the mean of the data set, \(\bar{x}\).
• Compute the squared deviations of each data point from the mean.
• Average these squared deviations (this is the variance).
• Take the square root of the variance to get the standard deviation.

The formula is given by: \[s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\]Here, \(x_i\) represents each data point and \(n\) is the number of data points.Understanding the sample standard deviation gives insight into the variability of data, which is crucial for defining intervals, like confidence intervals, accurately.