Question

Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.

Answer 1

Explanation:

Hello!

Text from Exercise 1:

"The waking times (in minutes past 5:00 a.m.) of 40 people who start work at 8:00 a.m. are shown in the table at the left. Assume the population standard deviation is 45 minutes."

The study variable is X: waking time (past 5:00 a.m.) of one person, measured in minutes.

Assuming the variable has a normal distribution, the margin of error, d, of the CI for the population mean waking time is:

d=
`Z_(1-\alpha /2)` * (δ/√n)

You have to calculate the sample size for a 95% CI, σ= 45min and d=10 min.

For this you have to clear the sample size from the formula:

d=
`Z_(1-\alpha /2)` * (δ/√n)

`(d)/(Z_(1-\alpha /2))`=δ/√n

√n*
`(d)/(Z_(1-\alpha /2))`= δ

√n= δ *
`(Z_(1-\alpha /2))/(d)`

n= ( δ *
`(Z_(1-\alpha /2))/(d)`)²

n= (45*
`(1.96)/(10)`)²

n= 77.79≅ 78

You need a sample of at least 78 people to estimate the population mean of the waking time using a 95% CI.

I hope this helps!