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 300 trials, the touch therapists were correct 144 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?
0.5 (Type an integer or a decimal. Do not round.)
b. Using Emily's sample results, what is the best point estimate of the therapists' success rate?
0.48 (Round to three decimal places as needed.)
c. Using Emily's sample results, construct a 95\% confidence interval estimate of the proportion of correct responses made by touch therapists.
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Solution
Step 1 ：Given that Emily used a coin toss to select either her right hand or her left hand, the probability of guessing correctly by chance is 0.5. This is because a coin toss has two equally likely outcomes: heads or tails, which we can associate with Emily's right hand or her left hand.
Step 2 ：The best point estimate of the therapists' success rate is the proportion of correct responses in Emily's sample. This can be calculated as the number of correct responses divided by the total number of trials. In this case, the success rate is \(\frac{144}{300} = 0.48\).
Step 3 ：To construct a 95% confidence interval for the proportion of correct responses, we can use the formula for the confidence interval for a proportion: \(\hat{p} \pm Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, Z is the Z-score for the desired level of confidence (for 95% confidence, Z is approximately 1.96), and n is the sample size. The lower and upper bounds of the confidence interval are then \(\hat{p} - Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) and \(\hat{p} + Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), respectively.
Step 4 ：Substituting the given values into the formula, we get the lower bound as \(0.48 - 1.96\sqrt{\frac{0.48(1-0.48)}{300}} = 0.423\) and the upper bound as \(0.48 + 1.96\sqrt{\frac{0.48(1-0.48)}{300}} = 0.537\).
Step 5 ：\(\boxed{\text{Final Answer:}}\) a. The expected proportion of correct responses if the touch therapists made random guesses is 0.5. b. The best point estimate of the therapists' success rate is 0.48. c. The 95% confidence interval estimate of the proportion of correct responses made by touch therapists is approximately (0.423, 0.537).
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Solution
Step 1 ：Given that Emily used a coin toss to select either her right hand or her left hand, the probability of guessing correctly by chance is 0.5. This is because a coin toss has two equally likely outcomes: heads or tails, which we can associate with Emily's right hand or her left hand.
Step 2 ：The best point estimate of the therapists' success rate is the proportion of correct responses in Emily's sample. This can be calculated as the number of correct responses divided by the total number of trials. In this case, the success rate is \(\frac{144}{300} = 0.48\).
Step 3 ：To construct a 95% confidence interval for the proportion of correct responses, we can use the formula for the confidence interval for a proportion: \(\hat{p} \pm Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, Z is the Z-score for the desired level of confidence (for 95% confidence, Z is approximately 1.96), and n is the sample size. The lower and upper bounds of the confidence interval are then \(\hat{p} - Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) and \(\hat{p} + Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), respectively.
Step 4 ：Substituting the given values into the formula, we get the lower bound as \(0.48 - 1.96\sqrt{\frac{0.48(1-0.48)}{300}} = 0.423\) and the upper bound as \(0.48 + 1.96\sqrt{\frac{0.48(1-0.48)}{300}} = 0.537\).
Step 5 ：\(\boxed{\text{Final Answer:}}\) a. The expected proportion of correct responses if the touch therapists made random guesses is 0.5. b. The best point estimate of the therapists' success rate is 0.48. c. The 95% confidence interval estimate of the proportion of correct responses made by touch therapists is approximately (0.423, 0.537).
