Question

Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.25cm and a standard deviation of 0.43cm. Using the empirical rule, what percentage of the apples have diameters that are less than 6.82cm? Please do not round your answer.

Answer 1

16% of the apples have diameters that are less than 6.82cm

Explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) 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 = 7.25cm

Standard deviation = 0.43cm

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

Using the empirical rule, what percentage of the apples have diameters that are less than 6.82cm?

6.82 = 7.25 - 1*0.43

So 6.82 is one standard deviation below the mean.

Of the 50% of the diameters below the mean, 68% are within 1 standard deviation of the mean.

So 100-68 = 32% are not within 1 standard deviation of the mean.

0.5*0.32 = 0.16

16% of the apples have diameters that are less than 6.82cm