The distribution of heights of 6-year-old girls is approximately normally distributed with a mean of 46.0 inchesand a standard deviation of 2.7 inches. Aliyaah is 6 years old, and her height is 0.96 standard deviation above themean. Her friend Jayne is also 6 years old and is at the 93rd percentile of the height distribution. At whatpercentile is Aliyaah’s height, and how does her height compare to Jayne’s height?(A) Aliyaah’s height is at the 17th percentile of the distribution, and she is shorter than Jayne.(B) Aliyaah’s height is at the 67th percentile of the distribution, and she is shorter than Jayne.(C) Aliyaah’s height is at the 67th percentile of the distribution, and she is taller than Jayne.(D) Aliyaah’s height is at the 83rd percentile of the distribution, and she is shorter than Jayne.(E) Aliyaah’s height is at the 83rd percentile of the distribution, and she is taller than Jayne
Question
The distribution of heights of 6-year-old girls is approximately normally distributed with a mean of 46.0 inchesand a standard deviation of 2.7 inches. Aliyaah is 6 years old, and her height is 0.96 standard deviation above themean. Her friend Jayne is also 6 years old and is at the 93rd percentile of the height distribution. At whatpercentile is Aliyaah’s height, and how does her height compare to Jayne’s height?(A) Aliyaah’s height is at the 17th percentile of the distribution, and she is shorter than Jayne.(B) Aliyaah’s height is at the 67th percentile of the distribution, and she is shorter than Jayne.(C) Aliyaah’s height is at the 67th percentile of the distribution, and she is taller than Jayne.(D) Aliyaah’s height is at the 83rd percentile of the distribution, and she is shorter than Jayne.(E) Aliyaah’s height is at the 83rd percentile of the distribution, and she is taller than Jayne
Solution
To answer this question, we first need to understand what it means when it says Aliyaah's height is 0.96 standard deviation above the mean. In a normal distribution, a score that is 1 standard deviation above the mean corresponds to approximately the 84th percentile. Since 0.96 is very close to 1, we can estimate that Aliyaah's height is close to the 84th percentile.
Next, we compare Aliyaah's height to Jayne's. Since Jayne is at the 93rd percentile, she is taller than about 93% of 6-year-old girls. Since Aliyaah's height is close to the 84th percentile, she is taller than about 84% of 6-year-old girls. Therefore, Jayne is taller than Aliyaah.
So, the correct answer is (D) Aliyaah’s height is at the 83rd percentile of the distribution, and she is shorter than Jayne.
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