Question

Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of μ=22.5 in. and a standard deviation of σ=1.1 in. These data are often used in the design of different​ seats, including aircraft​ seats, train​ seats, theater​ seats, and classroom seats. Instead of using 0.05 for identifying significant​ values, use the criteria that a value x is significantly high if​ P(x or ​greater) ≤0.01 and a value is significantly low if​ P(x or ​less) ≤0.01.
Find the​ back-to-knee lengths separating significant values from those that are not significant.

Answer 1

Answer:

Measures equal or lower than 19.94 inches are significantly low.

Measures equal or higher than 25.06 inches are significantly high.

Step-by-step explanation:

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

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

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 22.5, \sigma = 1.1$

Find the​ back-to-knee lengths separating significant values from those that are not significant.

Significantly low

In this exercise, a value is going to be to significantly low if it has a pvalue of 0.01 or less. So we have to find X when Z has a pvalue of 0.01. This is between $Z = -2.32$ and $Z = -2.33$, so we use $Z = -2.325$

$Z = \frac{X - \mu}{\sigma}$

$-2.325 = \frac{X - 22.5}{1.1}$

$X - 22.5 = -2.325*1.1$

$X = 19.94$

Measures equal or lower than 19.94 inches are significantly low.

Significantly high

In this exercise, a value is going to be to significantly high if it has a pvalue of 0.99 or more. So we have to find X when Z has a pvalue of 0.99. This is $Z = 2.325$. So:

$Z = \frac{X - \mu}{\sigma}$

$2.325 = \frac{X - 22.5}{1.1}$

$X - 22.5 = 2.325*1.1$

$X = 25.06$

Measures equal or higher than 25.06 inches are significantly high.