Question

25 men from Pinellas County were randomly drawn from a population of 100,000 men and weighed. The average weight of a man from the sample was found to be 150 pounds with a standard deviation of 54 pounds, Assuming the survey follows a normal distribution, find the 95% confidence interval for the true mean weight of men.

Answer 1

Answer:

95% confidence interval for the true mean weight of men is [127.71 pounds , 172.30 pounds].

Step-by-step explanation:

We are given that the average weight of a man from the sample was found to be 150 pounds with a standard deviation of 54 pounds

25 men from Pinellas County were randomly drawn from a population of 100,000 men.

Firstly, the pivotal quantity for 95% 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 weight of a man = 150 pounds

            s  = sample standard deviation = 54 pounds

            n = sample of men = 25

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 true mean, $\mu$ is ;

P(-2.064 < $t_2_4$ < 2.064) = 0.95  {As the critical value of t at 24 degree of

                                        freedom are -2.064 & 2.064 with P = 2.5%}  

P(-2.064 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 2.064) = 0.95

P( $-2.064 \times }{\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.064 \times }{\frac{s}{\sqrt{n} } }$ ) = 0.95

P( $\bar X-2.064 \times }{\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.064 \times }{\frac{s}{\sqrt{n} } }$ ) = 0.95

95% confidence interval for $\mu$ = [ $\bar X-2.064 \times }{\frac{s}{\sqrt{n} } }$ , $\bar X+2.064 \times }{\frac{s}{\sqrt{n} } }$ ]

                                            = [ $150-2.064 \times }{\frac{54}{\sqrt{25} } }$ , $150+2.064 \times }{\frac{54}{\sqrt{25} } }$ ]

                                            = [127.71 pounds , 172.30 pounds]

Therefore, 95% confidence interval for the true mean weight of men is [127.71 pounds , 172.30 pounds].