Answer :
Answer:
a) The standard deviation of x must be 0.25 inches.
b) We need a sample of 125 to reduce the standard deviation of x⎯⎯⎯ to the value you found in part(a).
Step-by-step explanation:
The 68-95-99.7 states that:
68% percent of the measures of a normally distributed sample are within 1 standard deviation of the mean.
95% percent of the measures of a normally distributed sample are within 2 standard deviations of the mean.
99.7% percent of the measures of a normally distributed sample are within 3 standard deviations of the mean.
The standard deviation of the population is 2.8. This means that [tex]\sigma = 2.8[/tex].
(a) What standard deviation must x⎯⎯⎯ have so that 95% of all samples give an x⎯⎯⎯ within one-half inch of μ?
We want to have a sample in which 2 standard deviations are within 0.5 inches of the mean.
So, the standard deviation of the sample must be:
[tex]2s = 0.5[/tex]
[tex]s = 0.25[/tex]
The standard deviation of x must be 0.25 inches.
(b) How large an SRS do you need to reduce the standard deviation of x⎯⎯⎯ to the value you found in part(a)?
We have that the standard deviation of a sample of length n is given by the following formula:
[tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
We want [tex]s = 0.25[/tex] and we have [tex]\sigma = 2.8[/tex]. So
[tex]0.25 = \frac{2.8}{\sqrt{n}}[/tex]
[tex]0.25\sqrt{n} = 2.8[/tex]
[tex]\sqrt{n} = 11.2[/tex]
[tex]\sqrt{n}^{2} = (11.2)^{2}[/tex]
[tex]n = 125.44[/tex]
We need a sample of 125 to reduce the standard deviation of x⎯⎯⎯ to the value you found in part(a).
A) The standard deviation in which 95% of all samples give an x⎯⎯⎯ within one-half inch of μ is; s = 0.25
B) The size of the SRS needed to reduce the standard deviation of x⎯⎯⎯ to the value you found in part a is; n = 125 male students
We are given;
Population Standard deviation; σ = 2.8 inches
Now, from the 68-95-99.7 rule, we know that;
- At Confidence level of 68% of a normally distributed sample, the measures of the normally distributed sample are within 1 standard deviation of the mean.
- At 95% confidence level, the measures of a normally distributed sample are within 2 standard deviations of the mean.
- At 99.7% confidence level, the measures of a normally distributed sample are within 3 standard deviations of the mean.
A) We want a standard deviation in which 95% of all samples give an x⎯⎯⎯ within one-half inch of μ is given by;
2s = 0.5
I used 2s because at 95% confidence level, the measures of a normally distributed sample are within 2 standard deviations of the mean.
Thus; s = 0.5/2
s = 0.25
B) we want to find large an SRS do you need to reduce the standard deviation of x⎯⎯⎯ to the value you found in part a above is gotten from the formula;
s = σ/√n
where n is is the sample size;
0.25 = 2.8/√n
Rearrange to get;
√n = 2.8/0.25
n ≈ 125
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