Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.42cm and a standard deviation of 0.36cm. Using the empirical rule, what percentage of the apples have diameters that are between 7.06cm and 7.78cm

68% of the values are within 1 standard deviation of the mean. i.e. between μ - 1σ and μ + 1σ
95% of the values are within 2 standard deviations of the mean. i.e. between μ - 2σ and μ + 2σ
99.7% of the values are within 3 standard deviation of the mean. i.e. between μ - 3σ and μ + 3σ

So, we first need to find how many standard deviations away are the given two data points. This can be done by converting them to z-score. A z score tells us that how far is a data value from the mean. The formula to calculate the z-score is:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

x = 7.06 converted to z score will be:

[tex]z=\frac{7.06-7.42}{0.36}=-1[/tex]

x = 7.78 converted to z score will be:

[tex]z=\frac{7.78-7.42}{0.36}=1[/tex]

This means the two given values are 1 standard deviation away from the mean and we have to find what percentage of values are within 1 standard deviation of the mean.

From the first listed point of empirical formula, we can say that 68% of the data values lie within 1 standard deviation of the mean. Therefore, 68% of the diameters are between 7.06 cm and 7.78 cm