Answer:
The percentage of the women have size shoes that are greater than
9.94 is 16%
Step-by-step explanation:
* <em>Lets revise the empirical rule</em>
- The Empirical Rule states that almost all data lies within 3 standard
deviations of the mean for a normal distribution.
- 68% of the data falls within one standard deviation.
- 95% of the data lies within two standard deviations.
- 99.7% of the data lies Within three standard deviations
* <em>The empirical rule shows that</em>
# 68% falls within the first standard deviation (µ ± σ)
# 95% within the first two standard deviations (µ ± 2σ)
# 99.7% within the first three standard deviations (µ ± 3σ).
* <em>Lets solve the problem</em>
- The shoe sizes of American women have a bell-shaped distribution
with a mean of 8.42 and a standard deviation of 1.52
∴ μ = 8.42
- The standard deviation is 1.52
∴ σ = 1.52
- <u><em>One standard deviation (µ ± σ):</em></u>
∵ (8.42 - 1.52) = 6.9
∵ (8.42 + 1.52) = 9.94
- <u><em>Two standard deviations (µ ± 2σ):</em></u>
∵ (8.42 - 2×1.52) = (8.42 - 3.04) = 5.38
∵ (8.42 + 2×1.52) = (8.42 + 3.04) = 11.46
- <u><em>Three standard deviations (µ ± 3σ): </em></u>
∵ (8.42 - 3×1.52) = (8.42 - 4.56) = 3.86
∵ (8.42 + 3×1.52) = (8.42 + 4.56) = 12.98
- <em>We need to find the percent of American women have shoe sizes </em>
<em> that are greater than 9.94</em>
∵ The empirical rule shows that 68% of the distribution lies
within one standard deviation in this case, from 6.9 to 9.94
∵ We need the percentage of greater than 9.94
- <em>That means we need the area under the cure which represents more</em>
<em> than one standard deviation (more than 68%)</em>
∵ The total area of the curve is 100% and the area within one standard
deviation is 68%
∴ The area greater than one standard deviation = (100 - 68)/2 = 16
∴ The percentage of the women have size shoes that are greater
than 9.94 is 16%
