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.

Question

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.

Answer 1

To find the mean and standard deviation of the sampling distribution of x, we can use the properties of the normal distribution.

The mean of the sampling distribution (μx) is equal to the mean of the population (μ), which is 12.4 percent.

The standard deviation of the sampling distribution (σx) is equal to the population standard deviation (σ) divided by the square root of the sample size (n).

σx = σ / √n
= 20.6% / √5

Now, to find the interval containing 95.44 percent of all possible sample mean returns, we need to calculate the margin of error and then construct a confidence interval around the sample mean.

The margin of error (ME) is given by multiplying the standard deviation of the sampling distribution (σx) by the appropriate Z-score, which corresponds to the desired level of confidence.

For a 95.44 percent confidence level, we need to find the Z-score that leaves 2.28 percent in the tails of the normal distribution. Since the normal distribution is symmetric, we can find the Z-score corresponding to 1.14 percent (half of 2.28 percent) in the right tail using a standard normal distribution table or calculator.

Using a standard normal distribution table, we find that the Z-score for 1.14 percent is approximately 2.18.

The margin of error is then calculated as:
ME = Z * σx
= 2.18 * (20.6% / √5)

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

Confidence interval = (x - ME, x + ME)

Note that x is the mean return of the sampled stocks, which is unknown.

So, to summarize:

Mean of sampling distribution (μx) = 12.4 percent
Standard deviation of sampling distribution (σx) = 20.6% / √5
Margin of error (ME) = 2.18 * (20.6% / √5)
Confidence interval = (x - ME, x + ME)

Please note that this explanation assumes that the population is normally distributed and that the random sample is representative of the population.