Question

Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of mu equals 23.3 in. and a standard deviation of sigma equals 1.2 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)less than or equals0.01 and a value is significantly low if​ P(x or ​less)less than or equals0.01. Find the​ back-to-knee lengths separating significant values from those that are not significant. Using these​ criteria, is a​ back-to-knee length of 25.6 in. significantly​ high?

Answer 1

The back-to-knee lengths separating significant values from those that are not significant are 20.2 in. and 26.4 in.

Using these​ criteria, a​ back-to-knee length of 25.6 in. is not significantly​ high, as is within the not-significant values.

Explanation:

We have a normal distribution for the sitting​ back-to-knee length with mean 23.3 in. and standard deviation 1.2 in.

We have to calculate the critical values that limit the 1% right tail and 1% left tail.

This can be done looking for a standard normal distribution table for the values that satisfy:

`P(z>z^*)=0.01\;\;\;\text{(for the right tail)}\\\\P(z<-z^*)=0.01\;\;\;\text{(for the left tail)}`

This values for z are z=±2.576.

Then, we can calculate the bounds for our sitting​ back-to-knee length distribution as:

`X_1=\mu+z_1\cdot\sigma=23.3+(-2.597)\cdot 1.2=23.3-3.1=20.2\\\\ X_2=\mu+z_2\cdot\sigma=23.3+(2.597)\cdot 1.2=23.3+3.1=26.4`