Answer:
a) ![\sigma = 0.167](https://tex.z-dn.net/?f=%5Csigma%20%3D%200.167)
b) We need a sample of at least 282 young men.
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)
This Zscore is how many standard deviations the value of the measure X 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.
(a) What standard deviation must x have so that 99.7% of allsamples give an x within one-half inch of μ?
To solve this problem, we use the 68-95-99.7 rule. This rule states that:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviations of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we want 99.7% of all samples give X within one-half inch of
. So
must have
and
must have
.
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)
![3 = \frac{0.5}{\sigma}](https://tex.z-dn.net/?f=3%20%3D%20%5Cfrac%7B0.5%7D%7B%5Csigma%7D)
![3\sigma = 0.5](https://tex.z-dn.net/?f=3%5Csigma%20%3D%200.5)
![\sigma = \frac{0.5}{3}](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%5Cfrac%7B0.5%7D%7B3%7D)
![\sigma = 0.167](https://tex.z-dn.net/?f=%5Csigma%20%3D%200.167)
(b) How large an SRS do you need to reduce the standard deviationof x to the value you found in part (a)?
You know from government data that heights of young men are approximately Normal with standard deviation about 2.8 inches. This means that ![\sigma = 2.8](https://tex.z-dn.net/?f=%5Csigma%20%3D%202.8)
The standard deviation of a sample of n young man is given by the following formula
![s = \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
We want to have ![s = 0.167](https://tex.z-dn.net/?f=s%20%3D%200.167)
![0.167 = \frac{2.8}{\sqrt{n}}](https://tex.z-dn.net/?f=0.167%20%3D%20%5Cfrac%7B2.8%7D%7B%5Csqrt%7Bn%7D%7D)
![0.167\sqrt{n} = 2.8](https://tex.z-dn.net/?f=0.167%5Csqrt%7Bn%7D%20%3D%202.8)
![\sqrt{n} = \frac{2.8}{0.167}](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%20%3D%20%5Cfrac%7B2.8%7D%7B0.167%7D)
![\sqrt{n} = 16.77](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%20%3D%2016.77)
![\sqrt{n}^{2} = 16.77^{2}](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%5E%7B2%7D%20%3D%2016.77%5E%7B2%7D)
![n = 281.23](https://tex.z-dn.net/?f=n%20%3D%20281.23)
We need a sample of at least 282 young men.