Question

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 323 ​trials, the touch therapists were correct 153 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? (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? (Round to three decimal places as​ needed.)

c. Using​ Emily's sample​ results, construct a 90​% confidence interval estimate of the proportion of correct responses made by touch therapists. ___< p <___ (Round to three decimal places as​ needed.)

d. What do the results suggest about the ability of touch therapists to select the correct hand by sensing energy​ fields?

Answer 1

Solution:

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? (Type an integer or a decimal. Do not​ round.)

Answer: ½ or 0.5

If the touch therapists made random guesses, then they would have only two choices for selection either right or left hand. So, proportion of correct responses would be expected as ½ or 0.5.

b. Using​ Emily's sample​ results, what is the best point estimate of the​ therapists' success​ rate? (Round to three decimal places as​ needed.)

We are given

Number of successes = X = 153

Number of trials = n = 323

Point estimate of the therapists’ success rate = X/n = 153/323 = 0.473684

Required point estimate = 0.474

c. Using​ Emily's sample​ results, construct a 90​% confidence interval estimate of the proportion of correct responses made by touch therapists. ___< p <___ (Round to three decimal places as​ needed.)

Confidence interval for Population Proportion

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We have

x = 153

n = 323

P = x/n = 153/323 = 0.473684211

Confidence level = 90%

Critical Z value = 1.6449

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.473684211 ± 1.6449* sqrt(0.473684211*(1 – 0.473684211)/323)

Confidence Interval = 0.473684211 ± 1.6449*0.0278

Confidence Interval = 0.473684211 ± 0.0457

Lower limit = 0.473684211 - 0.0457 = 0.4280

Upper limit = 0.473684211 + 0.0457 = 0.5194

Confidence interval = (0.428, 0.519)

0.428 < p < 0.519

d. What do the results suggest about the ability of touch therapists to select the correct hand by sensing energy​ fields?

We are 90% confident that the ability of touch therapists to select the correct hand by sensing energy fields will be lies between 42.8% and 51.9%.