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).


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?

Verified step by step guidance

1. Step 1: Understand the problem. Emily conducted an experiment where touch therapists guessed which hand she selected. The problem asks us to determine the expected proportion of correct responses if the therapists were guessing randomly.
2. Step 2: Recall the concept of probability for random guesses. If the therapists are guessing randomly, there are two equally likely outcomes: guessing the correct hand or the incorrect hand. The probability of guessing correctly is therefore 1/2.
3. Step 3: Express the expected proportion mathematically. The expected proportion of correct responses, denoted as p, is equal to the probability of guessing correctly, which is p = 1/2.
4. Step 4: Relate this to the context of the problem. Since the therapists are making random guesses, the expected proportion of correct responses is based solely on chance, and no skill or ability to sense energy fields is involved.
5. Step 5: Conclude that the expected proportion of correct responses under random guessing is 0.5, or 50%.

Key Concepts

Proportion and Probability

Proportion refers to the part of a whole, expressed as a fraction or percentage. In this context, if touch therapists were guessing randomly between two options (right or left hand), we would expect a 50% success rate, as there are two equally likely choices. This concept is fundamental for understanding the baseline expectation against which the therapists' performance can be compared.

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 (e.g., 95%). In this exercise, constructing a confidence interval for the proportion of correct responses will help assess whether the therapists' performance significantly differs from random guessing, providing insight into the effectiveness of touch therapy.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, one might set up a null hypothesis stating that the touch therapists' success rate is equal to random guessing (50%). By comparing the observed proportion of correct guesses to this expected value, we can determine if the results are statistically significant or likely due to chance.