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March 20th, 2007, 05:52 AM   #1
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Confidence Intervals

f1) A study was conducted to estimate the mean amount spent on birthday
gifts for a typical family having two children. A sample of 150 was taken,
and the mean amount spent was $225. Assuming a standard deviation equal to
$50, find the 95% confidence interval for the population mean, the mean for
all such families.

2) What sample size would be needed to estimate the population mean to
within one-half standard deviation with 95% confidence?

.
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March 29th, 2007, 08:59 PM   #2
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1)

Given

n = 150

sample mean = y bar = 225

assuming the stdev of 50 given is the population stdev and not the sample stdev, i.e. sigma = 50

100(1-a)=95% so a = 5% (alpha)

z_(a/2) = z_0.025 = 1.96

Then the 95% CI for mu is:
( y bar - z_a/2 * sigma/ n^(1/2) , y bar + z_a/2 * sigma/ n^(1/2) )
= ( 225 - 1.96 * 50/150^(1/2) , 225 + 1.96 * 50/150^(1/2) )
= ( 217.00, 233.00)



2)

From before the 95% CI for mu is:
( y bar - z_a/2 * sigma/ n^(1/2) , y bar + z_a/2 * sigma/ n^(1/2) )

one half stdev = 1/2 * 50 = 25
so we want 2 * (z_a/2 * sigma/ n^(1/2)) <25>= (2*1.96*50/25)^2 = 61.4656

So the sample size needed is at least 62.






note: I may be wrong.
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