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 stock and calculating the mean return , x, of the sample stocks. Find the mean and the standard deviation of the sampling distribution of x, and find an interval containing 95.44 of all possible sample 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 stock and calculating the mean return , x, of the sample stocks. Find the mean and the standard deviation of the sampling distribution of x, and find an interval containing 95.44 of all possible sample 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 is equal to the mean of the population, which is 12.4 percent in this case. The standard deviation of the sampling distribution, often referred to as the standard error, is equal to the standard deviation of the population divided by the square root of the sample size. Let's calculate it using the given values:

Standard Deviation of Sampling Distribution (Standard Error) = 20.6 / √(5)
= 9.213 percent

Therefore, the mean of the sampling distribution is 12.4 percent, and the standard deviation (standard error) is 9.213 percent.

Now, let's find the interval containing 95.44 percent of all possible sample returns. Since the sampling distribution follows a normal distribution, we can use the Z-score table to find the corresponding Z-value.

The Z-value for a given percentage (95.44 percent in this case) represents the number of standard deviations from the mean. We can find this Z-value by finding the cumulative probability in the Z-score table that corresponds to the desired percentage.

The percentage of interest is 95.44 percent, which is (100 - 95.44) / 2 = 2.28 percent in each tail of the distribution. In the Z-score table, this value corresponds to a Z-value of approximately 1.96.

Let's calculate the interval using the mean, standard deviation (standard error), and the Z-value:

Lower Limit = Mean - (Z-value * Standard Error)
= 12.4 - (1.96 * 9.213)
= -6.22 percent

Upper Limit = Mean + (Z-value * Standard Error)
= 12.4 + (1.96 * 9.213)
= 30.26 percent

Therefore, the interval containing 95.44 percent of all possible sample returns is approximately -6.22 percent to 30.26 percent.