Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Touch TherapyWhen she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily’s hand without seeing it

and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in “A Close Look at Therapeutic Touch,”Journal of the American Medical Association,Vol. 279, No. 13).

a.Given that Emily used a coin toss to select either her right hand or her left hand, what proportion of correct responses would be expected if the touch therapists made random guesses?

b.Using Emily’s sample results, what is the best point estimate of the therapists’ success rate?

c.Using Emily’s sample results, construct a 99% confidence interval estimate of the proportion of correct responses made by touch therapists.

d.What do the results suggest about the ability of touch therapists to select the correct hand by sensing an energy field?

In a sample of 280 trials, the touch therapists were correct 123 times.

a.

There are two possible outcomes to select: right hand or left hand

Since one of the two hands has the coin, the number of correct outcomes is equal to 1.

The proportion of correct responses is computed below:

$$\begin{gathered} p=\frac{\text{Number}\text{of}\text{correct}\text{responses}}{\text{Total}\text{number}\text{of}\text{responses}} \\ =\frac{1}{2} \\ =0.5 \end{gathered}$$

Thus, the proportion of correct responses is equal to 0.5.

b.

The best point estimate of the therapists’ success rate is the sample proportion of correct guesses made by the therapists.

Let x be the number of correct responses.

Let n be the sample size.

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{123}{280} \\ =0.439 \end{gathered}$$

The best point estimate of therapists’ success rate is 0.439.

c.

The formula for the confidence interval is given as follows:

$CI=\hat{p}-E<p<\hat{p}+E$

The right-tailed critical value of $z_{\frac{\alpha }{2}}$when $\alpha =0.01$is equal to 2.5758.

The value of the margin of error (E) is computed below:

$$\begin{gathered} E=z_{\frac{\alpha }{2}}\times \sqrt{\frac{\hat{p}\hat{q}}{n}} \\ =z_{\frac{0.01}{2}}\times \sqrt{\frac{0.44\times 0.56}{280}} \\ =2.5758\times \sqrt{\frac{0.44\times 0.56}{280}} \\ =0.0764 \end{gathered}$$

Substituting the required values, the following confidence interval is obtained:

$$\begin{gathered} \hat{p}-E<p<\hat{p}+E \\ 0.439-0.0764<p<0.439+0.0764 \\ 0.363<p<0.516 \end{gathered}$$

Thus, the 99% confidence interval estimate of the proportion of correct responses is equal to (0.36, 0.516).

d.

The sample proportion of correct guesses made by the therapists is equal to 0.439, which is even lower than 0.5.

Moreover, the confidence interval suggests that the proportion of correct responses is quite low.

Therefore, touch therapists are not very efficient in their ability to sense the energy field.