Chapter 23, Problem 10E

(a)

To determine

To explain carefully what the software input means.

Explanation of Solution

It is given that data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl. It is as:

  $\begin{array}{l}t\text{-Inter val for }\mu \\ 29.202<\mu <31.844\\ \mu =\text{ Age}\\ \text{95\% Confidence interval}\end{array}$

This means that we are $\text{95\%}$ confident that the average age at which babies begin to crawl is between $29.202\text{ weeks and 31}\text{.844 weeks}$ .

(b)

To determine

To find out what is the margin of error for this interval.

Answer to Problem 10E

The margin of error is $1.321$ weeks.

Explanation of Solution

It is given that data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl. It is as:

  $\begin{array}{l}t\text{-Inter val for }\mu \\ 29.202<\mu <31.844\\ \mu =\text{ Age}\\ \text{95\% Confidence interval}\end{array}$

The margin of error can be calculated as:

  $\begin{array}{l}\text{Margin of error}=\frac{31.844-29.202}{2}\\ =\frac{2.642}{2}\\ =1.321\end{array}$

Thus, the margin of error is $1.321$ weeks.

(c)

To determine

To explain what would the margin of error be larger or smaller, if the researcher has calculated a $90\%$ confidence interval.

Answer to Problem 10E

It will be smaller.

Explanation of Solution

It is given that data collected by child development scientists produced this confidence interval for the average age (in weeks) at which babies begin to crawl. It is as:

  $\begin{array}{l}t\text{-Inter val for }\mu \\ 29.202<\mu <31.844\\ \mu =\text{ Age}\\ \text{95\% Confidence interval}\end{array}$

And the margin of error is $1.321$ weeks. Now, if the researcher has calculated a $90\%$ confidence interval then the confidence level is decreased, then we become less sure that the confidence interval contains the true population mean and thus the confidence interval would become narrower.

If the confidence interval becomes narrower, then the margin of error decreases, because the margin of error is half the width of the confidence interval.