In 2003, the average stock price for companies making up the S&P 500 was $30, and thestandard deviation was $8.20 (BusinessWeek, Special Annual Issue, Spring 2003). Assumethe stock prices are normally distributed.a. What is the probability that a company will have a stock price of at least $40?b. What is the probability that a company will have a stock price no higher than $20?c. How high does a stock price have to be to put a company in the top 10%?

a. To find the probability that a company will have a stock price of at least $40, we need to calculate the z-score first. The z-score is calculated by subtracting the mean from the desired value and dividing it by the standard deviation. In this case, the mean is$30 and the standard deviation is $8.20.

z = (40 - 30) / 8.20 z = 1.22

Next, we need to find the probability corresponding to this z-score using a standard normal distribution table or a calculator. The probability of a z-score of 1.22 or higher can be found as 1 minus the cumulative probability up to 1.22.

P(Z ≥ 1.22) = 1 - P(Z ≤ 1.22)

Using a standard normal distribution table or calculator, we find that P(Z ≤ 1.22) is approximately 0.8888. Therefore,

P(Z ≥ 1.22) = 1 - 0.8888 P(Z ≥ 1.22) ≈ 0.1112

So, the probability that a company will have a stock price of at least $40 is approximately 0.1112 or 11.12%.

b. To find the probability that a company will have a stock price no higher than $20, we follow a similar process. First, we calculate the z-score:

z = (20 - 30) / 8.20 z = -1.22

Next, we find the probability corresponding to this z-score:

P(Z ≤ -1.22)

Using a standard normal distribution table or calculator, we find that P(Z ≤ -1.22) is approximately 0.1112.

So, the probability that a company will have a stock price no higher than $20 is approximately 0.1112 or 11.12%.

c. To determine how high a stock price needs to be to put a company in the top 10%, we need to find the z-score corresponding to the top 10% of the distribution. This is equivalent to finding the z-score that has a cumulative probability of 0.90.

Using a standard normal distribution table or calculator, we find that the z-score corresponding to a cumulative probability of 0.90 is approximately 1.28.

Now, we can calculate the stock price using the z-score formula:

z = (X - 30) / 8.20 1.28 = (X - 30) / 8.20

Solving for X, we have:

X - 30 = 1.28 * 8.20 X - 30 = 10.536 X ≈ 40.536

Therefore, a stock price of approximately $40.536 would put a company in the top 10%.