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

Confidence intervals
ANSWERED
asked 2021-08-12

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 358 ​trials, the touch therapists were correct 172 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 \(\displaystyle{90}​\%\) confidence interval estimate of the proportion of correct responses made by touch therapists.
Round to three decimal places as​ needed - ?\(\displaystyle{<}{p}{<}\)?

Expert Answers (1)

2021-08-13

Step 1
Given:
\(\displaystyle{n}={358}\)
\(\displaystyle{x}={172}\)
a) The proportion of correct responses is obtained as below:
\(\displaystyle\hat{{{p}}}={\frac{{{x}}}{{{n}}}}\)
\(\displaystyle={\frac{{{172}}}{{{358}}}}\)
\(\displaystyle={0.480}\)
Thus, the proportion of correct responses is 0.480.
Step 2
b) The best point estimate of the​ therapists' success​ rate is obtained as below:
Point estimate of \(\displaystyle{p}=\hat{{{p}}}\)
\(\displaystyle={0.480}\)
Thus, the best point estimate of the​ therapists' success​ rate is 0.480.
c) From the Standard Normal Table, the value of \(\displaystyle{z}\times\) for \(\displaystyle{90}\%\) level is \(\displaystyle{1.645}\).
The 90​% confidence interval estimate of the proportion of correct responses made by touch therapist is obtained as below:
sample statistic \(\displaystyle\pm{z}\times{S}{E}=\hat{{{p}}}\pm{z}\times\sqrt{{{\frac{{\hat{{{p}}}{\left({1}-\hat{{{p}}}\right)}}}{{{n}}}}}}\)
\(\displaystyle={0.480}\pm{1.645}\times\sqrt{{{\frac{{{0.480}\times{0.520}}}{{{358}}}}}}\)
\(\displaystyle={0.480}\pm{\left({1.645}\times{0.0264}\right)}\)
\(\displaystyle={0.480}\pm{0.0434}\)
\(\displaystyle={\left({0.437},\ {0.523}\right)}\)
Thus, the \(\displaystyle{90}​\%\) confidence interval estimate of the proportion of correct responses made by touch therapist is \(\displaystyle{0.437}{<}{p}{<}{0.523}\).

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