Question

You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign​ sedan, you decide to estimate the resale value of this car with a 99​% confidence interval. You manage to obtain data on 17 recently resold​ 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $ 12 comma 740 with a standard deviation of $ 800. What is the 99​% confidence interval for the true mean resale value of a​ 5-year-old car of this​ model?

Answer 1

99% confidence interval for the true mean resale value of a​ 5-year-old car of this​ model is [$12,173.24 , $13,306.76].

Explanation:

We are given that you manage to obtain data on 17 recently resold​ 5-year-old foreign sedans of the same model.

These 17 cars were resold at an average price of $ 12 comma 740 with a standard deviation of $ 800.

Firstly, the pivotal quantity for 99% confidence interval for the true mean is given by;

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

where,
`\bar X` = sample average price = $12,740

s = sample standard deviation = $800

n = sample of cars = 17

`\mu` = true population mean

Here for constructing 99% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.

So, 99% confidence interval for the population mean,
`\mu` is ;

P(-2.921 <
`t_1_6` < 2.921) = 0.99 {As the critical value of t at 16 degree of

freedom are -2.921 & 2.921 with P = 0.5%}

P(-2.921 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 2.921) = 0.99

P(
`-2.921 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`2.921 * {(s)/(√(n) ) }` ) = 0.99

P(
`\bar X-2.921 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+2.921 * {(s)/(√(n) ) }` ) = 0.99

99% confidence interval for
`\mu` = [
`\bar X-2.921 * {(s)/(√(n) ) }` ,
`\bar X+2.921 * {(s)/(√(n) ) }` ]

= [
`12,740-2.921 * {(800)/(√(17) ) }` ,
`12,740+2.921 * {(800)/(√(17) ) }` ]

= [$12,173.24 , $13,306.76]

Therefore, 99% confidence interval for the true mean resale value of a​ 5-year-old car of this​ model is [$12,173.24 , $13,306.76].