Problem 6SE

Refer to Exercise 5. Additional pieces of yarn will be sampled in order to improve the precision of the confidence interval. Which would increase the precision the most: sampling 50 additional pieces of yarn from the old batch, 50 additional pieces from the new batch, or 25 additional pieces from each batch?

Answer:

Step 1 of 1:

In this question, we are asked to improve the precision of the confidence interval while referring to exercise 5.

Which would increase the precision the most:

sampling 50 additional pieces of yarn from the old batch, 50 additional pieces from the new batch, or 25 additional pieces from each batch.

and

Here and are variable.

Then 90% confidence interval for the difference is,

Width of confidence interval =

Case 1: sampling 50 additional pieces of yarn from the old batch.

Hence the sample sizes are and

The width of a confidence interval is , and the confidence interval is .

Case 2: sampling 25 additional pieces from each batch.

Hence the sample sizes are and

The width of a confidence interval is , and the confidence interval is .

Case 3: sampling 50 additional pieces of yarn from the new batch.

Hence the sample sizes are and

The width of a confidence interval is , and the confidence interval is .

So from the above cases, case (3) has the width of while other two case (1) and (2) and has the width of 0.416 and 0.378 respectively.

We know that the width of a confidence interval indicates the precision. Narrow confidence intervals show precise knowledge of parameters, while wide confidence interval shows imprecise knowledge.

Hence 50 additional pieces sampled from the new batch (case 3) will increase the precision of the confidence interval.