Critical Thinking. In Exercises 17–28, use the data and confid...

Question

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.

Accepted Answer

Hello. In this video, we are told that a para parapsychology researcher tested whether participants could remote view the color of a hidden card. On each trial, a participant guessed red or black for a face-down playing card's color. Over 250 independent trials participants were correct 110 times. We want to construct a 99% confidence interval for the true proportion P of correct guesses. OK, so in order to start this problem, the first thing we want to do is we want to go ahead and find our point estimate. Now, the point estimate in this case is going to be the amount of people that were correct over the 250 independent trials. So we are going to label the point estimate as Phat, and that is going to be 110/250, which simplifies to be 0.44. OK. Since we are trying to construct a confidence interval, we need to determine the margin of error and the critical values. So let's go ahead and start by determining the critical values. Now, we are going to assume that the data is normally distributed, and what this means is that we need to find the area of the tail ends of the normal distribution. This is going to be labeled as alpha, and alpha is equal to 1 minus our confidence level. Now, in this case here, we are trying to construct a 99% confidence interval, and that means that alpha is going to be 1 minus 0.99, which simplifies to be 0.01. Now, this is the area of the tail ends of the normal distribution, but since there are two tail ends, we have to divide this area by 2. So, 0.01 divided by 2, gives us a value of 0.005. So what we are trying to do is we are trying to find a Z score whose area contains 0.005. By using the tea table in the appendix, we get that the critical value that has an area of 0.005 is 2.576. This is going to be the critical value that we want to work with, and we will go ahead and use this to help us find the margin of error. But before we do that, we need to determine the standard error of the data. Now, the standard error for a point estimate is defined as the square root of the point estimate, multiplied by 1 minus the point estimate, divided by the amount of people in the trials. Now this is going to give us the square root of 0.44, multiplied by 1 minus 0.44, divided by 250, and with the help of a calculator, this is going to approximate to be 0.0314. Now that we have the standard error and the critical points, we can use this to find the margin of error. Now, the margin of error is defined as the Z score critical point multiplied by the standard error. This is going to be 2.576, multiplied by 0.0314. And again, with the help of a calculator, this is going to approximate to be 0.081. Finally, now that we have the margin of error, we are going to use this margin of error to find the endpoints of our confidence interval. Now, the end points of the confidence interval is going to be defined as the point estimate, plus or minus the margin of error. So, if we expand this and plug in the values, this is going to give us 0.44 minus 0.081 and 0.44 plus 0.081. This is going to simplify to be 0.359 and 0.521. And if we represent these decimals or these decimals as percentages, we get that P is between 52.1%. And 35.9%. So that means that the amount of people that participate in the trials and are correct, is between 35.9% and 52.1%. And with that being said, the solution to this problem is going to be D. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Proportion

Sample Size

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