Question

Suppose a normally distributed set of stock prices with 3500 observations has a mean of 101 and a standard deviation of 10. Use the 68-95-99.7 Rule to determine the number of observations in the data set expected to be between the values 91 and 121.

Answer 1

The number of observations in the data set expected to be between the values 91 and 121 is of 2853.

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 101, Standard deviation = 10

Percentage of of observations in the data set expected to be between the values 91 and 121.

91 = 101 - 10

So 91 is one standard deviation below the mean.

121 = 101 + 2*10

So 121 is two standard deviations above the mean

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

Of those 50% below the mean, 68% are between one standard deviation below the mean(91) and the mean(101).

Of those 50% above the mean, 95% are between the mean(101) and two standard deviations above the mean(121).

So the percentage of observations in this interval is of:

`P = 0.5*0.68 + 0.5*0.95 = 0.815`

Number of observations in the interval

81.5% of 3500. So

0.815*3500 = 2853

The number of observations in the data set expected to be between the values 91 and 121 is of 2853.