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Tatiana [17]
3 years ago
13

Suppose that the sitting​ back-to-knee length for a group of adults has a normal distribution with a mean of mu equals 24.4 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 26.6 in. significantly​ high?
Mathematics
1 answer:
solong [7]3 years ago
6 0

Answer:

P(X \geq 26.6) = 0.0336, which is greater than 0.01. So a back-to-knee length of 26.6 in. is not significantly high.

Step-by-step explanation:

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

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

Z = \frac{X - \mu}{\sigma}

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:

\mu = 24.4, \sigma = 1.2

In this problem, a value x is significantly high if:

P(X \geq x) = 0.01

Using these​ criteria, is a​ back-to-knee length of 26.6 in. significantly​ high?

We have to find the probability of the length being 26.6 in or more, which is 1 subtracted by the pvalue of Z when X = 26.6. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{26.6 - 24.4}{1.2}

Z = 1.83

Z = 1.83 has a pvalue of 0.9664.

1 - 0.9664 = 0.0336

P(X \geq 26.6) = 0.0336, which is greater than 0.01. So a back-to-knee length of 26.6 in. is not significantly high.

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