A large hotel chain is trying to decide whether to convert more of its rooms to non-smoking rooms.
In a random sample of 400 guests last year, 166 had requested non-smoking rooms. This year, 205
guests in a sample of 380 preferred the non-smoking rooms. Would you recommend that the hotel
chain convert more rooms to non-smoking? Support your recommendation by testing the appropriate
hypotheses at a 0.01 level of significance.
To determine whether the hotel chain should convert more rooms to non-smoking, we can perform a hypothesis test.
First, let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The proportion of guests who prefer non-smoking rooms has not changed (p1 = p2).
Alternative Hypothesis (Ha): The proportion of guests who prefer non-smoking rooms has increased (p1 < p2).
Next, we need to calculate the sample proportions for each year:
Sample proportion for last year (p̂1) = 166/400 ≈ 0.415
Sample proportion for this year (p̂2) = 205/380 ≈ 0.539
Now, we can use a hypothesis test to determine if the observed increase in the proportion of guests preferring non-smoking rooms is statistically significant.
Assuming the conditions for a hypothesis test are met (random sampling, normal approximations), we can use the z-test. The test statistic can be calculated as follows:
z = (p̂1 - p̂2) / √((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2))
where n1 and n2 are the sample sizes for last year and this year, respectively.
In this case, n1 = 400 and n2 = 380. Substituting the values, we get:
z = (0.415 - 0.539) / √((0.415 * (1 - 0.415) / 400) + (0.539 * (1 - 0.539) / 380))
Calculating this gives us z ≈ -4.32.
Now, we compare the test statistic to the critical value at a 0.01 level of significance. Since our alternative hypothesis is a one-tailed test (p1 < p2), we need to find the critical value for a left-tailed test at a significance level of 0.01.
Looking up the critical value in the z-table (or using a calculator), the critical value corresponding to a significance level of 0.01 is approximately -2.33.
Since the test statistic (-4.32) is less than the critical value (-2.33), we have enough evidence to reject the null hypothesis.
Therefore, we recommend that the hotel chain convert more rooms to non-smoking, as there is a statistically significant increase in the proportion of guests who prefer non-smoking rooms.