# Statistics Help?

Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.36 and a standard deviation of 1.46. Using the empirical rule, what percentage of American women have shoe sizes that are at least 11.28? Please do not round your answer.

Relevance

You need to find the z-score, which can be found with:

n = m + sz

where:

n = the data point you are looking for (11.28)

m = the mean (8.36)

s = the standard deviation (1.46)

z = the z-score (unknown)

Substitute what we know and solve:

n = m + sz

11.28 = 8.36 + 1.46z

2.92 = 1.46z

2 = z

The empirical rule states that 95% of data points are within 2 standard deviations from the mean. This would include the range of:

-2s < m < 2s

You are asked for "at least 11.28" which means you don't want the range above, but the upper portion of what's left.

So if we don't want the range from -2s < m < 2s, we look at the negative data points:

x < -2s and x > 2s

which takes up 5% of the data.

But since we only want the values above 2s, not below -2s, we throw out that half, which leaves us with:

2.5% of the data points being larger than 11.28 for a shoe size.

• z(x=11.28) = (11.28-8.36)/1.46 = 2

95/2 = 47.5

100-50-47.5 = 2.5