Question

Empirical research on stock market data indicates that over the course of a year, 40% of stocks go up.

A random sample of 600 stocks is going to be chosen at the beginning of next year. Let p-hat be the proportion of the stocks in the sample that go up over the course of a year.

A.) Find the mean of p-hat.
B.) Find the standard deviation of p-hat.
C.) Compute an approximation for P(p-hat < 0.42) , which is the probability that fewer than 42% of the stocks in the sample go up over the course of the year. Round your answer to four decimal places.

Answer 1

Mean of p-hat: 0.4 (40%)

The standard deviation of p-hat: 0.027

P(p-hat < 0.42): 0.7640 (approximately 76.40%)

How can you find the mean and Standard deviation of p-hat ?

The mean of p-hat is equal to the population proportion itself (assuming random sampling), which is:

μ p = p = 40%

B. The standard deviation of p-hat depends on the sample size (n) and the population proportion (p):

σ p = √(p * (1 - p) / n)

σ p= √(0.4 * 0.6 / 600) ≈ 0.027

C. P(p-hat < 0.42):

Since the sample size is large (n > 30), we can use the normal approximation. The z-score for p-hat = 0.42 is:

z = (p-hat - μ p) / σ p = (0.42 - 0.4) / 0.027 ≈ 0.74

Using a standard normal distribution table or calculator, we can find the probability P(z < 0.74) ≈ 0.7642.

Therefore, the approximate probability that fewer than 42% of the stocks in the sample go up over the course of a year is:

P(p-hat < 0.42) ≈ 0.7642

P(p-hat < 0.42) ≈ 0.7640