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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 .
(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:
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:
.
We want and we have
. So
We need a sample of 125 to reduce the standard deviation of x⎯⎯⎯ to the value you found in part(a).