Question

A sample of 16 elements is selected from a population with a reasonably symmetrical distribution. The sample mean is 100 and the sample standard deviation is 40. We can say that we are 90% confident that $LaTeX: \mu\:$ μis between what two numbers?

11. What is the lower number? Maintain three decimal points in your calculations and give at least two decimal points in your answer.

12. What is the upper number? Maintain three decimal points in your calculations and give at least two decimal points in your answer.

13. We want to estimate the population mean travel-to-work time for residents of San Francisco. Supposed that a preliminary simple random sample of residents in San Francisco is used to develop a planning value estimate of 10 minutes for the population standard deviation. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of one minute, what sample size should be used? Assume 95% confidence is required.

Answer 1

Solution :

Given that,

Point estimate = sample mean = $\bar x$ = 100

sample standard deviation = s = 40

sample size = n = 16

Degrees of freedom = df = n - 1 = 16 - 1 = 15

At 90% confidence level

$\alpha$ = 1 - 90%

$\alpha$ =1 - 0.90 =0.10

$\alpha$/2 = 0.05

t$\alpha$/2,df = t_{0.05, 15 =} 1.753

Margin of error = E = t_{$\alpha$/2,df} * (s /$\sqrt$n)

= 1.753 * ( 40/ $\sqrt$ 16)

Margin of error = E = 17.53

The 90% confidence interval estimate of the population mean is,

$\bar x$  ± E

= 100  ± 17.53

= ( 82.47, 117.53 )

11) lower limit = 82.47

12) upper limit = 117.53

13) Population standard deviation = $\sigma$ = 10

Margin of error = E = 1

At 95% confidence level the z is,

$\alpha$ = 1 - 95%

$\alpha$ = 1 - 0.95 = 0.05

$\alpha$/2 = 0.025

Z_$\alpha$/2 = 1.96

sample size = n = [Z_$\alpha$/2* $\sigma$ / E] ²

n = [ 1.96 * 10 / 1]²

n = 384.16

Sample size = n = 385