91影视

A sample of 12 adults is randomly selected. The actual population鈥檚 mean sleep time is equal to 6.0 hours. It is claimed that the mean sleep time is less than 7 hours.

The appropriate hypotheses for testing the given claim are written as follows:

Null hypothesis: The mean sleep time is equal to 7.0 hours.

\({H_0}:\mu = 7\)

Alternative Hypothesis: The mean sleep time is less than 7.0 hours.

\({H_0}:\mu < 7\)

The test is left-tailed.

The power of the test refers to the probability of rejecting the null hypothesis when it is actually false. In simple words, the value of the power of the test explains the probability of making the correct conclusion of the claim.

Here, the value of the power of the test is equal to 0.4943.

The actual population鈥檚 mean sleep time is equal to 6.0 hours. It is claimed that the mean sleep time is less than 7 hours.

Thus, it means that there is a 49.43% probability of making the correct conclusion that the mean sleep time is less than 7 hours (rejecting the false null hypothesis) when the population鈥檚 true mean sleep time is equal to 6.0 hours.

The value of\(\beta \)is computed using the following formula:

\(\begin{array}{c}{\rm{Power}} = 1 - \beta \\\beta = 1 - {\rm{Power}}\end{array}\)

Therefore, the value of\(\beta \)is equal to:

\(\begin{array}{c}\beta = 1 - {\rm{Power}}\\ = 1 - 0.4943\\ = 0.5057\end{array}\)

Thus, the value of \(\beta \) is equal to 0.5057.

The value of\(\beta \)is the value of the Type II error which is the probability of failing to reject the null hypothesis when it is actually false.

In simple words,\(\beta \)refers to the probability of making an incorrect conclusion about the stated claim.

For the given claim, the value of\(\beta \)represents the probability of rejecting the claim that the mean sleep time is less than 7 hours when the true population鈥檚 mean sleep time is actually equal to 6.0 hours.