Suppose that the sitting back-to-knee length for a group of adults has a normal distribution with a mean of μ=22.5 in. and a standard deviation of σ=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) ≤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.

Question

14-11-2019
Mathematics

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Suppose that the sitting back-to-knee length for a group of adults has a normal distribution with a mean of μ=22.5 in. and a standard deviation of σ=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) ≤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.

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joaobezerra joaobezerra · Answer 1 · 2019-11-14T17:33:36+01:00

Answer:

Measures equal or lower than 19.94 inches are significantly low.

Measures equal or higher than 25.06 inches are significantly high.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 22.5, \sigma = 1.1[/tex]

Find the back-to-knee lengths separating significant values from those that are not significant.

Significantly low

In this exercise, a value is going to be to significantly low if it has a pvalue of 0.01 or less. So we have to find X when Z has a pvalue of 0.01. This is between [tex]Z = -2.32[/tex] and [tex]Z = -2.33[/tex], so we use [tex]Z = -2.325[/tex]

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-2.325 = \frac{X - 22.5}{1.1}[/tex]

[tex]X - 22.5 = -2.325*1.1[/tex]

[tex]X = 19.94[/tex]

Measures equal or lower than 19.94 inches are significantly low.

Significantly high

In this exercise, a value is going to be to significantly high if it has a pvalue of 0.99 or more. So we have to find X when Z has a pvalue of 0.99. This is [tex]Z = 2.325[/tex]. So:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]2.325 = \frac{X - 22.5}{1.1}[/tex]

[tex]X - 22.5 = 2.325*1.1[/tex]

[tex]X = 25.06[/tex]

Measures equal or higher than 25.06 inches are significantly high.