Question

An industrial designer wants to determine the average amount of time an adults take to assemble an"easy to assemble"toy.a sample of 16 times yielded an average time of 19.92 minutes,with a sample standard deviation of 19.92 minutes,with a sample standard deviation of 5.73 minutes. assuming normality of assembly times, provide a 95% confidence interval for the mean assembly time and interpret the results.​

Answer 1

95% confidence interval for the mean assembly time is [16.87 minutes , 22.97 minutes].

Step-by-step explanation:

We are given that an industrial designer wants to determine the average amount of time an adults take to assemble an "easy to assemble" toy.

A sample of 16 times yielded an average time of 19.92 minutes, with a sample standard deviation of 5.73 minutes.

Firstly, the Pivotal quantity for 95% confidence interval for the population mean is given by;

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

where, $\bar X$ = sample average time = 19.92 minutes

            s = sample standard deviation = 5.73 minutes

            n = sample size = 16

            $\mu$ = population mean assembly time

Here for constructing 95% confidence interval we have used One-sample t-test statistics as we don't know about population standard deviation.

So, 95% confidence interval for the population mean, $\mu$ is ;

P(-2.131 < $t_1_5$ < 2.131) = 0.95  {As the critical value of t at 15 degree

                                          of freedom are -2.131 & 2.131 with P = 2.5%}  

P(-2.131 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 2.131) = 0.95

P( $-2.131 \times {\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.131 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

P( $\bar X-2.131 \times {\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.131 \times {\frac{s}{\sqrt{n} } }$ ) = 0.95

95% confidence interval for $\mu$ = [ $\bar X-2.131 \times {\frac{s}{\sqrt{n} } }$ , $\bar X+2.131 \times {\frac{s}{\sqrt{n} } }$ ]

                                      = [ $19.92-2.131 \times {\frac{5.73}{\sqrt{16} } }$ , $19.92+2.131 \times {\frac{5.73}{\sqrt{16} } }$ ]

                                      = [16.87 , 22.97]

Therefore, 95% confidence interval for the mean assembly time is [16.87 minutes , 22.97 minutes].

Now, the interpretation of the above confidence interval is that we are 95% confident that the mean assembly time will lie between 16.87 minutes and 22.97 minutes.