Question

A researcher records the repair cost for 2727 randomly selected VCRs. A sample mean of $86.26$ 86.26 and standard deviation of $11.77$ 11.77 are subsequently computed. Determine the 90%90% confidence interval for the mean repair cost for the VCRs. Assume the population is approximately normal.

Answer 1

Answer:

90% confidence interval for the mean repair cost for the VCR's is [82.396 , 90.124].

Step-by-step explanation:

We are given that a researcher records the repair cost for 27 randomly selected VCR's. A sample mean of $86.26 and standard deviation of $11.77 are subsequently computed.

Firstly, the pivotal quantity for 90% confidence interval for the true mean repair cost for the VCR's is given by;

         P.Q. = $\frac{\bar X - \mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, $\bar X$ = sample mean repair cost = $86.26

             s = sample standard deviation = $11.77

             n = sample of VCR's = 27

             $\mu$ = true mean repair cost

Here for constructing 90% confidence interval we have used t statistics because we don't know about population standard deviation.

So, 90% confidence interval for the true mean repair cost, $\mu$ is ;

P(-1.706 < $t_2_6$ < 1.706) = 0.90  {As the critical value of t at 26 degree of

                                                 freedom are -1.706 & 1.706 with P = 5%}

P(-1.706 < $\frac{\bar X - \mu}{\frac{s}{\sqrt{n} } }$ < 1.706) = 0.90

P( $-1.706 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X - \mu}$ < $1.706 \times {\frac{s}{\sqrt{n} } }$ ) = 0.90

P( $\bar X -1.706 \times {\frac{s}{\sqrt{n} }$ < $\mu$ < $\bar X +1.706 \times {\frac{s}{\sqrt{n} }$ ) = 0.90

90% confidence interval for $\mu$ = [ $\bar X -1.706 \times {\frac{s}{\sqrt{n} }$ , $\bar X +1.706 \times {\frac{s}{\sqrt{n} }$ ]

                                                 = [ $86.26 -1.706 \times {\frac{11.77}{\sqrt{27} }$ , $86.26 +1.706 \times {\frac{11.77}{\sqrt{27} }$ ]

                                                 = [82.396 , 90.124]

Therefore, 90% confidence interval for the true mean repair cost for the VCR's is [82.396 , 90.124].