An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will she need to be 95% confident that her sample mean will be within 15 seconds of the true mean? Assume that it is known from previous studies that σ = 40 seconds.

So, the required sample size is , that she will need to be 95% confident that her sample mean will be within 15 seconds of the true mean.

Step-by-step explanation:

Given that,

Standard deviation () = 40

and we have to find how large a sample is needed to be 95% confident that her sample mean will be within 15 seconds of the true mean.

Now,

If is used as an estimate of , we can be % confident that the error will not exceed a specified amount when the sample size is

             ........ (i)

                              where is the z value leaving an area of to the right.

So,

      % = 95%

We have    and , Now using equation (i), the required sample size is,

             ........(ii)

Now we have to find the value of .

So, the is the z-value leaving an area of 0.025 to the right {the area left of the is (1-0.025) = 0.975.}

Using Normal Probability table, we see that the closest z-value which leaving an area of 0.025 to the right (i.e. an area of 0.975 to the left) is

Now,

Using equation (ii),

                                   =

                                   ≈

So, the required sample size is