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
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 = 22.5, \sigma = 1.1](https://tex.z-dn.net/?f=%5Cmu%20%3D%2022.5%2C%20%5Csigma%20%3D%201.1)
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
and
, so we use ![Z = -2.325](https://tex.z-dn.net/?f=Z%20%3D%20-2.325)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-2.325 = \frac{X - 22.5}{1.1}](https://tex.z-dn.net/?f=-2.325%20%3D%20%5Cfrac%7BX%20-%2022.5%7D%7B1.1%7D)
![X - 22.5 = -2.325*1.1](https://tex.z-dn.net/?f=X%20-%2022.5%20%3D%20-2.325%2A1.1)
![X = 19.94](https://tex.z-dn.net/?f=X%20%3D%2019.94)
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
. So:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![2.325 = \frac{X - 22.5}{1.1}](https://tex.z-dn.net/?f=2.325%20%3D%20%5Cfrac%7BX%20-%2022.5%7D%7B1.1%7D)
![X - 22.5 = 2.325*1.1](https://tex.z-dn.net/?f=X%20-%2022.5%20%3D%202.325%2A1.1)
![X = 25.06](https://tex.z-dn.net/?f=X%20%3D%2025.06)
Measures equal or higher than 25.06 inches are significantly high.