Answer :
Answer:
a) [tex]\sigma = 0.167[/tex]
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 [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]
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 [tex]\mu[/tex]. So [tex]X - \mu = 0.5[/tex] must have [tex]Z = 3[/tex] and [tex]X - \mu = -0.5[/tex] must have [tex]Z = -3[/tex].
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]3 = \frac{0.5}{\sigma}[/tex]
[tex]3\sigma = 0.5[/tex]
[tex]\sigma = \frac{0.5}{3}[/tex]
[tex]\sigma = 0.167[/tex]
(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 [tex]\sigma = 2.8[/tex]
The standard deviation of a sample of n young man is given by the following formula
[tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
We want to have [tex]s = 0.167[/tex]
[tex]0.167 = \frac{2.8}{\sqrt{n}}[/tex]
[tex]0.167\sqrt{n} = 2.8[/tex]
[tex]\sqrt{n} = \frac{2.8}{0.167}[/tex]
[tex]\sqrt{n} = 16.77[/tex]
[tex]\sqrt{n}^{2} = 16.77^{2}[/tex]
[tex]n = 281.23[/tex]
We need a sample of at least 282 young men.