Question

Oscar-Winning Actresses Data Set 14 “Oscar Winner Age” in Appendix B lists ages of actresses when they won Oscars, and the summary statistics are n = 87, x = 36.2 years, and s = 11.5 years. Use a 0.01 significance level to test the claim that the mean age of actresses when they win Oscars is greater than 30 years.

Short answer

The null hypothesis is rejected, which means that there is sufficient evidence to support the claim that the mean age of the actresses was greater than 30 years when they won the Oscars.

Step-by-step solution

Given information

The statistics for the Oscar winners’ 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.

State the hypotheses

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,

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

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

Compute the test statistic

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.

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

Compute the critical value

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.

State the decision

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 \({H_0}\).

Conclusion

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.