Question

Find and interpret a 95% confidence interval to estimate the average number of bolts per box for all boxes in the population. Round to 3 decimal places.

Answer 1

Complete Question

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Answer:

The 95% confidence interval is  $49.85 < \mu < 54.15$

This means that there is 95% chance that the true population mean is within this interval

Step-by-step explanation:

From the question we are told that

    The  sample size is n = 30  

    The sample mean is  $\= x = 52$

    The population standard deviation is  $\sigma = 6$

Given that the confidence level is 95% then the level of confidence is evaluate as

             $\alpha = 100 - 95$

            $\alpha = 5\%$

            $\alpha = 0.05$

Next we obtain the critical value of  $\frac{\alpha }{2}$ from the normal distribution table , the values is

                 $Z_{\frac{\alpha }{2} } = 1.96$

Generally the margin of error is mathematically represented as

           $E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }$

substituting values

            $E = 1.96 * \frac{ 6 }{\sqrt{30} }$

           $E = 2.147$

The  95% confidence interval is mathematically represented as

          $\= x - E < \mu < \= x + E$

substituting values

         $52 - 2.147 < \mu < 52 + 2.147$

         $49.85 < \mu < 54.15$