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?
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.
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