# Free Solved Statistics Problems

*The Wall Street Journal *reported that almost all the major stock market indexes had posted strong gains in the last 12 months. The one-year return for the S&P 500, a group of 500 very large companies, was approximately + 27%. The one-year return in the Russell 2000, a group of 2000 small companies, was approximately + 52%. Historically, the one-year returns are approximately normal. The standard deviation in the S&P 500 returns is approximately 20%, and in the Russell 2000 the standard deviation is approximately 35%.

a. What is the probability that a stock in the S&P 500 gained 30% or more in the last year? Gained 60% or more in the last year?

b. What is the probability that a stock in the S&P 500 lost money in the last year? Lost 30% or more?

c. Repeat (a) and (b) for a stock in the Russell 2000.

d. Write a short summary on your findings. Be sure to include a discussion of the risks associated with a large standard deviation.

**Solution: a.-) **We’ll analyze S&P 500. We know we can consider the mean earnings of the S&P 500 to be

_{}

(measured in %). Also we know that the approximate standard deviation for the S&P 500 is

_{}

(also measured in %). We call again_{}to the return of a stock in the last year. We are looking for the probability that_{}, and the probability that_{}as in the previous part. We have that

_{}

But _{} has a standard normal distribution. Therefore:

_{}

The last probability is found using the EXCEL function NORMSDIST. In the same vein

_{}

Again, _{} has a standard normal distribution. Therefore:

_{}

Again we used the function NORMSDIST.

**b.-) **We are looking for the probability that_{}now.

_{}

Similarly, to get the probability that_{}we do:

_{}

**c.-) **We’ll analyze Russell 2000 now. We know we can consider the mean earnings of the Russell 2000 to be

_{}

(measured in %). Also we know that the approximate standard deviation for the S&P 500 is

_{}

(also measured in %). Let’s call _{}to the return of a stock in the last year. We are looking for the probability that_{}, and the probability that_{}. We have that

_{}

On the other hand

_{}

**d.-) **In both S&P 500 and Russell 2000, the gains are very strong, but due to their relatively high standard deviation (20% and 35%), the variability could be high as well. That means that even though the gains are +27% for S&P 500, still there is 8% probability that a stock lost money last year. Despite of the variability, the probability of good returns are still very high for a stock. For example, with a 73.52% probability a Russell 2000 stock gained 30% or more.

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