Question

Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of mu equals 23.4 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?

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Answer 1

$Let~X= the ~sitting~back~to~knee~length~for~a~group~of~adults\sim N(23.4,~1.2^2).\\ c~is~significantly~high~if~P(X\geq c)=0.01~or,~\Phi((c-23.4)/1.2)=0.99\\ =\Phi( 2.3263).\\ c=23.4+1.2* 2.3263=26.19\\ k~is~significantly~low~if~P(X\leq k)=0.01~or,\\ ~\Phi(-(k-23.4)/1.2)=\Phi( 2.3263)\Rightarrow k=23.4-1.2* 2.3263=20.61\\$

$Since~20.61<25.6<26.19~so ~a~back-to-knee~ length ~of ~25.6 ~in. ~is~not~\\ significantly~high.$