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RanH · Answer 1 · 2021-07-31T06:50:01+0000

Answer:

a) 11.51% probability that a random college woman has a height of 68 inches or more

b) This height is 70.135 inches.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

The normal distribution has two parameters, which are the mean and the standard deviation.

In a set with mean
$\mu$ and standard deviation
$\sigma$ , the zscore of a measure X is given by:

$Z = (X - \mu)/(\sigma)$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 65, \sigma = 2.5$

a. What is the probability that a random college woman has a height of 68 inches or more?

This is 1 subtracted by the pvalue of Z when X = 68. So

$Z = (X - \mu)/(\sigma)$

Z = (68 - 65)/(2.5)

Z = 1.2

Z = 1.2 has a pvalue of 0.8849

1 - 0.8849 = 0.1151

11.51% probability that a random college woman has a height of 68 inches or more

b. To be in the Tall Club, a woman must have a height such that only 2% of women are taller. What is this height?

This weight is the 100-2 = 98th percentile, which is the value of X when Z has a pvalue of 0.98. So X when Z = 2.054.

$Z = (X - \mu)/(\sigma)$

2.054 = (X - 65)/(2.5)

X - 65 = 2.054*2.5

X = 70.135

This height is 70.135 inches.

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