Questions: In a science fair project, Emily conducted an 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 332 trials, the touch therapists were correct 159 times. Complete parts (a) through (d). 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 90% confidence interval estimate of the proportion of correct responses made by touch therapists.

In a science fair project, Emily conducted an 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 332 trials, the touch therapists were correct 159 times. Complete parts (a) through (d). 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 90% confidence interval estimate of the proportion of correct responses made by touch therapists.
Transcript text: In a science fair project, Emily conducted an 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 332 trials, the touch therapists were correct 159 times. Complete parts (a) through (d). 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 90% confidence interval estimate of the proportion of correct responses made by touch therapists.
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Solution

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Solution Steps

Step 1: Expected Proportion of Correct Responses

Given that Emily used a coin toss to select either her right hand or her left hand, the expected proportion of correct responses if the therapists made random guesses is:

\[ p_0 = 0.5 \]

Step 2: Best Point Estimate of Success Rate

Using Emily's sample results, the best point estimate of the therapists' success rate is calculated as follows:

\[ \hat{p} = \frac{x}{n} = \frac{159}{332} \approx 0.479 \]

Step 3: Confidence Interval for the Proportion of Correct Responses

To construct a 90% confidence interval for the proportion of correct responses, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:

  • \(\hat{p} = 0.479\)
  • \(n = 332\)
  • \(z\) for a 90% confidence level is approximately \(1.645\).

Calculating the confidence interval:

\[ 0.479 \pm 1.645 \cdot \sqrt{\frac{0.479(1 - 0.479)}{332}} \approx 0.434 \text{ to } 0.524 \]

Thus, the 90% confidence interval for the proportion of correct responses is:

\[ (0.434, 0.524) \]

Final Answer

  • Expected proportion of correct responses: \(0.5\)
  • Best point estimate of success rate: \(0.479\)
  • 90% confidence interval for the proportion of correct responses: \((0.434, 0.524)\)

\[ \boxed{(0.434, 0.524)} \]

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