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 Therapy When 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).




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

Verified step by step guidance

1. Step 1: Identify the sample proportion (p̂) using the formula p̂ = x / n, where x is the number of correct responses (123) and n is the total number of trials (280). This gives the proportion of correct responses observed in the sample.
2. Step 2: Determine the standard error (SE) of the sample proportion using the formula SE = sqrt((p̂ * (1 - p̂)) / n). This measures the variability of the sample proportion.
3. Step 3: Find the critical value (z) for a 99% confidence level. For a 99% confidence interval, the z-value corresponds to the area under the standard normal curve, which is approximately 2.576.
4. Step 4: Calculate the margin of error (ME) using the formula ME = z * SE. This represents the range of error around the sample proportion.
5. Step 5: Construct the confidence interval for the population proportion (p) using the formula: Confidence Interval = p̂ ± ME. This provides the range within which the true proportion of correct responses is likely to fall with 99% confidence.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if the same sampling method were repeated multiple times, approximately 99% of the calculated intervals would contain the true proportion. This concept is crucial for estimating the reliability of sample data in making inferences about a larger population.

Proportion

In statistics, a proportion is a type of ratio that represents the part of a whole. It is calculated by dividing the number of successful outcomes by the total number of trials. In the context of Emily Rosa's experiment, the proportion of correct responses by touch therapists is determined by dividing the number of correct identifications (123) by the total trials (280), providing insight into their performance relative to chance.

Sample Size

Sample size refers to the number of observations or trials included in a statistical sample. A larger sample size generally leads to more reliable estimates and narrower confidence intervals, as it reduces the margin of error. In Emily's study, the sample size of 280 trials is significant, as it allows for a more accurate estimation of the proportion of correct responses and enhances the validity of the confidence interval constructed.