Question

A survey asked college freshman how far away from their hometown is from the college campus. A random sample of 36 students is taken with a mean distance of 100 miles and a standard deviation of 30 Complete parts (a) and (b) below (a) Find the 95% confidence interval for the population mean Lower bound Upper bound (Round your answer to one decimal place) (b) Suppose you want the estimate to be within 3 miles of the population mean. Determine the minimum sample size needed to construct a 95% confidence interval for the population mean under this condition Minimum sample size - Your answer to a whole number

Answer 1

a)

Mean (T) = 100 Sample size (n) = 36 Standard deviation (s) = 30 Confidence interval (in %) = 95

z @ 95.0% = 1.96

$\\\text{Since we know that} \\\\\\\text{Confidence interval = }\bar{x} \pm z\frac{s}{\sqrt{n}} \\\text{Required confidence interval = }(100.0-1.96\frac{30.0}{\sqrt{36}}, 100.0+1.96\frac{30.0}{\sqrt{36}})$

Required confidence interval = (100.0-1.96(5.0), 100.0+1.96(5.0))

Required confidence interval = (100.0 - 9.8, 100.0 + 9.8)

Required confidence interval = (90.2, 109.8)

Lower bound = 90.2

Upper bound = 109.8

b)

Margin of error (e) = 3

Since we know that

S n Margin of error = 2 30.0 3.0 = 1.96- Vn 30.0 Vn = 1.965 3.0 Vm= 19.6 n= 384.16 = 384

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