Question

Suppose a normally distributed set of data with 4800 observations has a mean of 195 and a standard deviation of 12. Use the 68-95-99.7 Rule to determine the number of observations in the data set expected to be below a value of 231. Round your result to the nearest single observation.

Answer 1

The number of observations in the data set expected to be below a value of 231 is of 4793.

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 = 195

Standard deviation = 12

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

Proportion of observations below 231:

231 = 195 + 3*12

So 231 is three standard deviations above the mean.

Of the 50% of observations below the mean, all are below 231.

Of the 50% of observations above the mean, 99.7% are between the mean of 195 and three standard deviations above the mean(231).

So, the proportion of observations below 231 is:

`P = 0.5*1 + 0.5*0.997 = 0.9985`

Out of 4800:

0.9985*4800 = 4793

The number of observations in the data set expected to be below a value of 231 is of 4793.