Question

Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of 24.5in. and a standard deviation of 1.1in. 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. Using these​ criteria, is a​ back-to-knee length of 26.7in. significantly​ high?

Answer 1

We have

$0.01=P\left( X\geqslant x \right) =P\left( \frac{X-24.5}{1.1}\geqslant \frac{x-24.5}{1.1} \right) =1-\varPhi \left( \frac{x-24.5}{1.1} \right) \Rightarrow \\\Rightarrow \frac{x-24.5}{1.1}=z_{0.99}\Rightarrow x=1.1z_{0.99}+24.5=1.1\cdot 2.3263+24.5=27.0589\\0.01=P\left( X\leqslant x \right) =P\left( \frac{X-24.5}{1.1}\leqslant \frac{x-24.5}{1.1} \right) =\varPhi \left( \frac{x-24.5}{1.1} \right) \Rightarrow \\\Rightarrow \frac{x-24.5}{1.1}=z_{0.01}\Rightarrow x=1.1z_{0.01}+24.5=1.1\cdot \left( -2.3263 \right) +24.5=21.9411$

Thus the length greater than 27.0589 is significantly high, the length less than 21.9411 is significantly low.

The length 26.7 is not significantly high.