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.
- llafferLv 76 months agoFavorite Answer
You need to find the z-score, which can be found with:
n = m + sz
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.
- Mike GLv 76 months ago
z(x=11.28) = (11.28-8.36)/1.46 = 2
95/2 = 47.5
100-50-47.5 = 2.5