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 \(\displaystyle{0.437}{<}{p}{<}{0.523}\).