Question

Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.23cm and a standard deviation of 0.35cm. Using the empirical rule, what percentage of the apples have diameters that are no more than 6.18cm? Please do not round your answer.

Answer 1

0.15% of the apples have diameters that are no more than 6.18cm.

Explanation:

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

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

Approximately 95% of the measures are within 2 standard deviations of the mean.

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

In this problem, we have that:

Mean = 7.23, standard deviation = 0.35.

What percentage of the apples have diameters that are no more than 6.18cm?

6.18 = 7.23 - 3*0.35

So 6.18cm is three standard deviations below the mean.

99.7% of the measures are within 3 standard deviations of the mean. The normal distribution is symmetric, which means that (100-99.7)/2 = 0.15% is 3 standard deviations or more below the mean, and 0.15% is 3 standard deviations or more above the mean.

Thus, 0.15% of the apples have diameters that are no more than 6.18cm.