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

Answer:

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].

Step-by-step 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. =  $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$  ~ $t_n_-_1$

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 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 2.921) = 0.99

P( $-2.921 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.921 \times {\frac{s}{\sqrt{n} } }$ ) = 0.99

P( $\bar X-2.921 \times {\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.921 \times {\frac{s}{\sqrt{n} } }$ ) = 0.99

99% confidence interval for $\mu$ = [ $\bar X-2.921 \times {\frac{s}{\sqrt{n} } }$ , $\bar X+2.921 \times {\frac{s}{\sqrt{n} } }$ ]

                                   = [ $12,740-2.921 \times {\frac{800}{\sqrt{17} } }$ , $12,740+2.921 \times {\frac{800}{\sqrt{17} } }$ ]

                                   = [$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].