Question

You work for a consumer advocate agency and want to find the mean repair cost of a washing machine. As part of your study, you randomly select 30 repair costs and find the mean to be $100.00. The standard deviation of the sample is $25.20. Calculate a 90% confidence interval for the population mean.

Answer 1

90% confidence interval for the population mean is [92.18 , 107.82].

Explanation:

We are given that as part of your study, you randomly select 30 repair costs and find the mean to be $100.00. The standard deviation of the sample is $25.20.

So, the pivotal quantity for 90% confidence interval for the population mean is given by;

P.Q. =
`(\bar X - \mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean = $100

s = sample standard deviation = $25.20

n = sample size = 30

`\mu` = population mean

So, 90% confidence interval for the population mean,
`\mu` is ;

P(-1.699 <
`t_2_9` < 1.699) = 0.90

P(-1.699 <
`(\bar X - \mu)/((s)/(√(n) ) )` < 1.699) = 0.90

P(
`-1.699 * {(s)/(√(n) )` <
`{\bar X - \mu}` <
`1.699 * {(s)/(√(n) )` ) = 0.90

P(
`\bar X - 1.699 * {(s)/(√(n) )` <
`\mu` <
`\bar X + 1.699 * {(s)/(√(n) )` ) = 0.90

90% confidence interval for
`\mu` = [
`\bar X - 1.699 * {(s)/(√(n) )` ,
`\bar X + 1.699 * {(s)/(√(n) )` ]

= [
`100 - 1.699 * {(25.20)/(√(30) )` ,
`100 + 1.699 * {(25.20)/(√(30) )` ]

= [92.18 , 107.82]

Therefore, 90% confidence interval for the population mean is [92.18 , 107.82].

Answer 2

90% confidence interval for the population mean is between a lower limit of $92.18 and an upper limit of $107.82.

Explanation:

Confidence interval for a population mean is given as mean +/- margin of error (E)

mean = $100

sd = $25.20

n = 30

degree of freedom = n-1 = 30-1 = 29

confidence level (C) = 90% = 0.9

significance level = 1 - C = 1 - 0.9 = 0.1 = 10%

critical value (t) corresponding to 29 degrees of freedom and 10% significance level is 1.699

E = t×sd/√n = 1.699×25.20/√30 = $7.82

Lower limit of population mean = mean - E = 100 - 7.82 = $92.18

Upper limit of population mean = mean + E = 100 + 7.82 = $107.82

90% confidence interval is ($92.18, $107.82)