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**1. Younger people cruises**

The data for this assignment are listed in the Excel file *Younger people cruises*.

The cruise ship business is rapidly increasing. Although cruises have long been associated with seniors, it now appears that younger people are choosing a cruise as their vacations. To determine whether this is true, an executive for a cruise line sampled passengers 2 years ago and this year and determined their ages.**a.** Formulate the null hypothesis and the alternative hypothesis .**b.** For the inference about comparing the population means of two independent populations (which is the case here) it is important to know whether the population variances are equal. Explain why. A test determines that the population variances may be assumed to be equal. Which test is used and explain why it comes to this conclusion?**c.** Do these data allow the executive to infer that cruise ships are attracting younger customers at a 1% significance level? Explain your answer.**d.** Estimate with 99% confidence the difference in ages between this year and 2 years ago.

**Solution.****a.** **b.** Applying the -test for the two samples, we find for the test statistic . At a 5% significance level, the critical value equals . Since we may infer that there is no evidence that the population variances differ. Thus we may apply the equal-variances test.**c.** Applying the -*test: two-sample assuming equal variances* (with the pooled variance estimator ) we find for the test statistic .

This is a one-sided test and thus the critical value is . The statistic is in the rejection region and thus there is not enough evidence to accept the null hypothesis. Thus, we may infer at a 1% significance level that cruise ships are attracting younger people. This result is also obtained by using Data Analysis, test.**d.** The 99% confidence interval is .

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