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 P/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). (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

Verify Given Statistics

We are given that the sample mean \( \bar{x} \approx 25.2 \) and the sample standard deviation \( s \approx 15.5 \). Using a calculator or relevant software, you input the 51 data points to calculate these statistics. Ensure they match approximately before proceeding with the confidence intervals.

02

Find 90% Confidence Interval

To find the 90% confidence interval for the mean \( \mu \), use the formula: \[ CI = \bar{x} \pm t \frac{s}{\sqrt{n}} \] where \( \bar{x} = 25.2 \), \( s = 15.5 \), and \( n = 51 \). The critical value \( t \) for 90% confidence for \( n-1 = 50 \) degrees of freedom can be found using t-distribution tables or software, yielding approximately \( t \approx 1.676 \). Substitute these into the formula to get the interval.

03

Calculate 90% Confidence Interval

Using the values from Step 2, the 90% confidence interval is calculated as follows: \[ CI = 25.2 \pm 1.676 \frac{15.5}{\sqrt{51}} \approx 25.2 \pm 3.64 \] This results in a confidence interval of approximately (21.56, 28.84).

04

Find 99% Confidence Interval

For the 99% confidence interval, use the similar formula: \( CI = \bar{x} \pm t \frac{s}{\sqrt{n}} \). The critical value \( t \) for 99% confidence with 50 degrees of freedom is approximately \( t \approx 2.678 \). Substitute these values into the formula.

05

Calculate 99% Confidence Interval

Using the values from Step 4, the 99% confidence interval is calculated as follows: \[ CI = 25.2 \pm 2.678 \frac{15.5}{\sqrt{51}} \approx 25.2 \pm 5.82 \] This results in a confidence interval of approximately (19.38, 31.02).

06

Interpretation of Confidence Intervals

Comparing the P/E ratios: Bank One with 12 is well below both intervals, suggesting it is likely a "value" stock. AT&T Wireless with 72 is well above both intervals, indicating it could be considered "overpriced." Disney with 24 falls within both intervals, suggesting it's typical relative to this sample of companies.

07

Check Requirements

According to the central limit theorem, since our sample size is greater than 30 (\( n = 51 \)), the sample mean will approximately follow a normal distribution regardless of the original distribution. Therefore, we do not need to assume a normal distribution of the \( P/E \) ratios.

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Key Concepts

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

Confidence Intervals

Confidence intervals offer a range of values within which we expect the population parameter, like a mean or proportion, to lie. They provide a way to understand the uncertainty surrounding our sample estimates.

• To create a confidence interval, we use the sample data mean and a margin of error.
• The margin of error is calculated using the sample standard deviation and a critical value from a statistical distribution like the t-distribution.

For example, in our exercise, we calculated a 90% confidence interval for the mean P/E ratio using sample mean 25.2 and critical value approximately 1.676. This interval indicates that we are 90% confident that the true mean P/E ratio lies within the interval calculated.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the sampling distribution of the sample mean will be approximately normally distributed, provided the sample size is large enough, generally more than 30.

• This theorem holds true regardless of the population's distribution.
• As the sample size increases, the distribution of the mean approaches a normal distribution.

In the context of our exercise, the sample size is 51. Thanks to the CLT, we can assume that the distribution of the sample mean P/E ratio approximates a normal distribution. Thus, allowing us to calculate confidence intervals safely without assuming that the P/E ratios themselves are normally distributed.

T-distribution

The t-distribution is a probability distribution that's used when working with small sample sizes or when the population standard deviation is unknown. In many ways, it's similar to a normal distribution but with heavier tails.

• The t-distribution becomes closer to a normal distribution as sample size increases.
• Critical values from the t-distribution are used in calculating confidence intervals when the sample size is small or the population standard deviation is unknown.

In our exercise, because the sample size is reasonable (51), we use the t-distribution instead of the normal distribution to determine the critical values. This provides more accurate intervals given that exact population parameters are unknown.

P/E Ratio

The P/E ratio, or Price-to-Earnings ratio, is a key financial metric used to evaluate the value of a company. It measures the current share price relative to its per-share earnings.

• Higher P/E ratios can indicate growth potential or possibly overvaluation.
• Conversely, lower ratios may suggest value stocks which are bargains according to market perceptions.

Our job as analysts is to interpret where a company stands relative to these ratios. In the case provided in the exercise, companies like Bank One and AT&T Wireless have significantly different P/E ratios, illustrating their perceived market valuations at the time.

Sample Standard Deviation

Sample standard deviation is a measure of the amount of variability or dispersion in a set of sample values. It gives us an idea of how much individual data points differ from the sample mean.

• A smaller standard deviation indicates data points are closer to the mean.
• Larger standard deviation suggests greater variability.

In our exercise, the P/E ratios showed a sample standard deviation of 15.5, reflecting the diversity in company valuations within the sample of 51 U.S. companies. This value is crucial when calculating confidence intervals as it determines the margin of error, helping us understand how varied the P/E ratios are from the mean.