Question

6.2.19-E Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of mu equals 24.1 in. and a standard deviation of sigma equals 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)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 26.2 in. significantly​ high? Find the​ back-to-knee lengths separating significant values from those that are not significant. ​Back-to-knee lengths greater than nothing in. and less than nothing in. are not​ significant, and values outside that range are considered significant.

Answer 1

Solution :

Let X be a random variable which represents the sitting​ back-to-knee length for a group of adults.

Given that, X ~ N(24.1, 1.1²)

μ = 24.1 inches and  σ = 1.1 inches

Let a back-to-knee length that separates the significant low length with the on significant lengths is k inches.

Hence, P(X ≤ k) = 0.01

We know that, if X ~ N(μ, σ²) then,

Using "qnorm" function of R we get, P(Z ≤ -2.326) = 0.01

Comparing, P(Z ≤ -2.326) = 0.01 and (1) we get,

Hence, a back-to-knee length less than or equal to 21.54 inches will be considered as significantly low.

Let a back-to-knee length that separates the significant high length with the non significant lengths is n inches.

Hence, P(X ≥ n) = 0.01

We know that, if X ~ N(μ, σ²) then,

Using "qnorm" function of R we get, P(Z ≥ 2.326) = 0.01

Comparing, P(Z ≥ 2.326) = 0.01 and (2) we get,

Hence, a back-to-knee length greater than or equal to 26.66 inches will be considered as significantly high.

The back-to-knee length of 26.2 inches is not significantly high.

​Back-to-knee lengths greater than 21.54 inches and less than 26.6- inches are not​ significant, and values outside that range are considered significant.

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