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Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99 ​% of all males.​ (Accommodating 100% of males would require very wide seats that would be much too​ expensive.) Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation of 0.9 in. Find Upper P 99 . That​ is, find the hip breadth for men that separates the smallest 99 ​% from the largest 1 ​%.

Respuesta :

Answer:

Upper P 99 = 16.49 in.

Step-by-step explanation:

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

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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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:

[tex]\mu = 14.4, \sigma = 0.9[/tex]

Find Upper P 99

This is X when Z has a pvalue of 0.99. So it is X when Z = 2.327.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]2.327 = \frac{X - 14.4}{0.9}[/tex]

[tex]X - 14.4 = 2.327*0.9[/tex]

[tex]X = 16.49[/tex]

Upper P 99 = 16.49 in.