Suppose that the percentage returns for a given year for all stocks listed on the New York Stock Exchange are approximately normally distributed with a mean of 12.4 percent and a standard deviation of 20.6 percent. Consider drawing a random sample of n= 5 stocks from the population of all stocks and calculating the mean return x, of the sampled stocks. Find the mean and the standard deviation of the sampling distribution of x, and find an interval containing 95.44 percent of all possible sample mean returns.

To find the mean and standard deviation of the sampling distribution of x, which is the mean return of a random sample of n=5 stocks, we can use the properties of sampling distributions.

The mean of the sampling distribution of x is the same as the population mean, which is 12.4 percent. This is because the sampling distribution of the mean is an unbiased estimator of the population mean.

The standard deviation of the sampling distribution of x, also known as the standard error, can be calculated using the formula:

standard error = population standard deviation / √(sample size)
= 20.6 percent / √5

Therefore, the standard error of the sampling distribution of x is approximately 9.19 percent.

To find the interval containing 95.44 percent of all possible sample mean returns, we can use the z-score and the cumulative distribution function (CDF) of the standard normal distribution.

First, we need to find the z-score corresponding to the desired area. The area of 95.44 percent corresponds to a z-score of 1.645. This can be found using a standard normal distribution table or calculator.

Next, we can calculate the margin of error, which is the product of the z-score and the standard error:

margin of error = z-score * standard error
= 1.645 * 9.19 percent

The margin of error is approximately 15.10 percent.

Finally, we can construct the interval by subtracting and adding the margin of error from the sample mean:

lower interval = sample mean - margin of error
= x - 15.10 percent

upper interval = sample mean + margin of error
= x + 15.10 percent

This interval will contain 95.44 percent of all possible sample mean returns. Note that the specific value for x will depend on the sample of stocks you actually draw.