91影视

Out of 280 trials, 123 guesses arecorrect by the touch therapists. It is claimed that that touch therapists randomly guess the answer.

The null hypothesis is written as follows.

The proportion of correct guesses is equal to 0.5.

\({H_0}:p = 0.5\).

The alternative hypothesis is written as follows.

The proportion of correct guesses is not equal to 0.5.

\({H_1}:p \ne 0.5\).

The test is two-tailed.

The sample size is n=280.

The sample proportion of correct guesses is computed below.

\[\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{correct}}\;{\rm{guesses}}}}{{{\rm{Sample}}\;{\rm{Size}}}}\\ = \frac{{123}}{{280}}\\ = 0.439\end{array}\].

The population proportion ofcorrect guesses is equal to 0.5.

The value of the test statistic is computed below.

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.439 - 0.5}}{{\sqrt {\frac{{0.5\left( {1 - 0.5} \right)}}{{280}}} }}\\ = - 2.032\end{array}\).

Thus, z=-2.032.

Referring to the standard normal table, the critical value of z at\(\alpha = 0.10\)for a two-tailed test is equal to 1.645.

Referring to the standard normal table, the p-value for the test statistic value of -2.032 is equal to 0.0422.

Asthe p-value is less than 0.10, the null hypothesis is rejected.

There is enough evidence to reject the claim that the touch therapists randomly guess the correct answer.

Asthe sample proportion of correct guesses equal to 43.9% is even less than 50% (as proposed by the method of random guesses), it can be said that touch therapists are not at all effective.