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 287 trials, the touch therapists were correct 135 times. Complete parts (a) through (d).
c. Using Emily's sample results, construct a 95\% confidence interval estimate of the proportion of correct responses made by touch therapists.
$
Solution
Step 1 ：Given that Emily conducted 287 trials and the touch therapists were correct 135 times, we can calculate the sample proportion (\(\hat{p}\)) as \(\frac{135}{287}\).
Step 2 ：Using a 95% confidence level, the corresponding z-score (z) is approximately 1.96.
Step 3 ：The formula for a confidence interval for a proportion is given by \(\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where n is the sample size.
Step 4 ：Substituting the given values into the formula, we get \(\hat{p} \pm 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{287}}\).
Step 5 ：Calculating the above expression, we get the 95% confidence interval estimate of the proportion of correct responses made by touch therapists as \((0.413, 0.528)\).
Step 6 ：Final Answer: The 95% confidence interval estimate of the proportion of correct responses made by touch therapists is \(\boxed{(0.413, 0.528)}\).
From Solvely APP
Solution
Step 1 ：Given that Emily conducted 287 trials and the touch therapists were correct 135 times, we can calculate the sample proportion (\(\hat{p}\)) as \(\frac{135}{287}\).
Step 2 ：Using a 95% confidence level, the corresponding z-score (z) is approximately 1.96.
Step 3 ：The formula for a confidence interval for a proportion is given by \(\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where n is the sample size.
Step 4 ：Substituting the given values into the formula, we get \(\hat{p} \pm 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{287}}\).
Step 5 ：Calculating the above expression, we get the 95% confidence interval estimate of the proportion of correct responses made by touch therapists as \((0.413, 0.528)\).
Step 6 ：Final Answer: The 95% confidence interval estimate of the proportion of correct responses made by touch therapists is \(\boxed{(0.413, 0.528)}\).
