Chapter 23, Problem 17E

(a)

To determine

To find a $95\%$ confidence interval for the true speed of light from these statistics.

Answer to Problem 17E

The confidence interval is $(709.895,802.545)$ .

Explanation of Solution

It is given that:

  $\begin{array}{l}n=23\\ \overline{x}=756.22\\ s=107.12\end{array}$

Now we will find the t -value by looking in the row starting with degrees of freedom. The degree of freedom is as:

  $\begin{array}{l}df=n-1\\ =23-1\\ =22\end{array}$

Thus, the t -value with $95\%$ confidence interval be:

  $t_{\alpha /2}=2.074$

The confidence interval is calculated as:

  $\begin{array}{l}\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =756.22-2.074\times \frac{107.12}{\sqrt{23}}\\ =709.895\\ \overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =756.22+2.074\times \frac{107.12}{\sqrt{23}}\\ =802.545\end{array}$

Thus, the confidence interval is $(709.895,802.545)$ .

(b)

To determine

To state in word what this interval means.

Answer to Problem 17E

We are $95\%$ confident that the true speed of light is between $299,709.895\text{Km/h and 299,802}\text{.545Km/h}$ , since all the data values had $299,000$ subtracted from them.

Explanation of Solution

It is given that:

  $\begin{array}{l}n=23\\ \overline{x}=756.22\\ s=107.12\end{array}$

And the confidence interval is $(709.895,802.545)$ . Thus, it means that we are $95\%$ confident that the true speed of light is between $299,709.895\text{Km/h and 299,802}\text{.545Km/h}$ , since all the data values had $299,000$ subtracted from them.

(c)

To determine

To find out what assumptions must you make in order to use your method.

Explanation of Solution

It is given that:

  $\begin{array}{l}n=23\\ \overline{x}=756.22\\ s=107.12\end{array}$

And the confidence interval is $(709.895,802.545)$ . The conditions to be checked are:

Random condition: It is satisfied because assuming that the measurements were randomly collected.

Independent condition: It is satisfied because assuming that measurements are independent.

Normal condition: It is satisfied because assuming that the population distribution of the measurements are normally distributed since the sample size is small.