### Understanding P/E Ratios and Confidence IntervalsThe concept of the Price-to-Earnings (P/E) ratio is crucial in evaluating stocks. This ratio is the price of a share of stock divided by the company’s estimated future earnings per share. Generally, high P/E ratios might indicate "growth" stocks, which could also be overpriced, while low P/E ratios may suggest "value" stocks or bargain stocks.#### Sample Data of P/E RatiosA random sample of 51 large companies in the United States provides the following P/E ratios:```11 35 19 13 15 21 40 18 60 72 9 2029 53 16 26 21 14 21 27 10 12 47 1433 14 18 17 20 19 13 25 23 27 5 168 49 44 20 27 8 19 12 31 67 51 2619 18 32```#### Calculations**(a) Calculating Sample Mean and Standard Deviation**Using a calculator, determine the sample mean ($ \bar{x} $) and sample standard deviation (s). Round your answers to four decimal places.- $ \bar{x} = \_\_\_\_\_\_ $- $ s = \_\_\_\_\_\_ $**(b) 90% Confidence Interval for P/E Population Mean**Find the 90% confidence interval for the P/E population mean $ \mu $ of all large U.S. companies. Round your answers to one decimal place.- Lower limit: \_\_\_\_\_\_- Upper limit: \_\_\_\_\_\_**(c) 99% Confidence Interval for P/E Population Mean**Calculate the 99% confidence interval for the P/E population mean $ \mu $. Round your answers to one decimal place.- Lower limit: \_\_\_\_\_\_- Upper limit: \_\_\_\_\_\_**(d) Evaluating Specific Companies’ P/E Ratios**Let’s consider P/E ratios for specific companies:

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The price of a share of stock divided by the 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 ratiost. 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 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to four decimal places.) X = S = (b) Find a 90% confidence interval for the P/E population mean u of all large U.S. companies. (Round your answers to one decimal place.) lower limit upper limit (c) Find a 99% confidence interval for the P/E population mean u of all large U.S. companies. (Round your answers to one decimal place.) lower limit upper limit

The price of a share of stock divided by the 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 ratiost. 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 (a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to four decimal places.) X = S = (b) Find a 90% confidence interval for the P/E population mean u of all large U.S. companies. (Round your answers to one decimal place.) lower limit upper limit (c) Find a 99% confidence interval for the P/E population mean u of all large U.S. companies. (Round your answers to one decimal place.) lower limit upper limit

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Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

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### Understanding P/E Ratios and Confidence Intervals

The concept of the Price-to-Earnings (P/E) ratio is crucial in evaluating stocks. This ratio is the price of a share of stock divided by the company’s estimated future earnings per share. Generally, high P/E ratios might indicate "growth" stocks, which could also be overpriced, while low P/E ratios may suggest "value" stocks or bargain stocks.

#### Sample Data of P/E Ratios

A random sample of 51 large companies in the United States provides the following P/E ratios:

```
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
```

#### Calculations

**(a) Calculating Sample Mean and Standard Deviation**

Using a calculator, determine the sample mean ($ \bar{x} $) and sample standard deviation (s). Round your answers to four decimal places.

- $ \bar{x} = \_\_\_\_\_\_ $
- $ s = \_\_\_\_\_\_ $

**(b) 90% Confidence Interval for P/E Population Mean**

Find the 90% confidence interval for the P/E population mean $ \mu $ of all large U.S. companies. Round your answers to one decimal place.

- Lower limit: \_\_\_\_\_\_
- Upper limit: \_\_\_\_\_\_

**(c) 99% Confidence Interval for P/E Population Mean**

Calculate the 99% confidence interval for the P/E population mean $ \mu $. Round your answers to one decimal place.

- Lower limit: \_\_\_\_\_\_
- Upper limit: \_\_\_\_\_\_

**(d) Evaluating Specific Companies’ P/E Ratios**

Let’s consider P/E ratios for specific companies:

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