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The statistics for the Oscar winners鈥� age is given as follows.

Sample size\(n = 87\).

Sample mean\(\bar x = 36.2\;{\rm{years}}\).

Sample standard deviation\(s = 11.5\;{\rm{years}}\).

Level of significance\(\alpha = 0.01\).

The claim states that the mean age of the winners is greater than 30 years.

Null hypothesis:The mean age of the actresses when they won the Oscars is equal to 30 years.

Alternative hypothesis:The mean age of the actresses when they won the Oscars is greater than 30 years.

Mathematically,

\({H_0}:\mu = 30\)

\({H_1}:\mu > 30\)

Assuming that the population is normally distributed and the sample is selected randomly, the t-test will apply if the standard deviation of the population is unknown.

The test statistic is given as follows.

\(\begin{array}{c}t = \frac{{\bar x - \mu }}{{\frac{s}{{\sqrt n }}}}\\ = \frac{{36.2 - 30}}{{\frac{{11.5}}{{\sqrt {87} }}}}\\ = 5.029\end{array}\).

The degree of freedom is computed as follows.

\(\begin{array}{c}df = n - 1\\ = 87 - 1\\ = 86\end{array}\).

Refer to the t-table for 86 degrees of freedom and the level of significance 0.01 for the one-tailed test for the critical value of 2.368.

If the critical value is greater than the test statistic, the null hypothesis is failed to be rejected. Otherwise, the null hypothesis is rejected.

The calculated test statistic value is greater than the critical value. So, reject \({H_0}\).

As the null hypothesis is rejected, there is enough evidence to support the claim that the mean age of all the Oscar-winning actresses is greater than 30 years.