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The distribution of heights of 6-year-old girls is approximately normally distributed with a mean of 46.0 inches and a standard deviation of 2.7 inches. Aaliyah is 6 years old, and her height is 0.96 standard deviation above the mean. Her friend Jayne is also 6 years old and is at the 93rd percentile of the height distribution. At what percentile is Aaliyah’s height, and how does her height compare to Jayne’s height?

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jennerfranco

Answer:

Aaliyah's height is the 84th percentile and the relationship that there exists between Aaliyah's height and between Jayne's height is that Aaliyah's height is less than Jayne's height.

Step-by-step explanation:

Below we can observe the empirical rule with a mean of 46.0 inches and a standard deviation of 2.7 inches. We have that 48.7 inches represents one standard deviation above the mean, so, we can consider that Aaliyah's height is the 84th percentile. On the other side, Jayne's height is the 93rd percentile of the height distribution. Therefore, the relationship that there exists between Aaliyah's height and between Jayne's height is that Aaliyah's height is less than Jayne's height.

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Using the standard normal distribution concept ;

  • Percentile of Aaliyah's height = 83rd percentile
  • Jayne is taller than Aaliyah

Writing out our parameters :

  • Mean, μ = 46.0 inches
  • Standard deviation, σ = 2.7 inches
  • Aaliyah's age = 0.96 standard deviation above the mean
  • Jayne's age = 93rd percentile

We need to use the data given to calculate the actual height of both Aaliyah and Jayne.

Aaliyah's actual height :

Actual height = μ + 0.96σ

Actual height = (46 + 0.96(2.7))

Aaliyah's actual height = 46 + 2.592 = 48.592

Using a normal distribution table :

Zscore of 0.96 corresponds to an area of about 0.83% = 83 percentile

Jayne's standardized age :

93rd percentile

P(x < Z) = 0.93

Using a normal distribution table, the Zscore value = 1.476

Using the Zscore relation to obtain the Jayne's actual height :

Zscore = (X - mean) / standard deviation

X = Jayne's actual height

1.476 = (X - 46) / 2.7

1.476 × 2.7 = X - 46

3.9852 = X - 46

X = 3.9852 + 46

X = 49.9852

Therefore, Comparing there actual height , we can conclude that Jayne is taller than Aaliyah

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