Cigarette Pack Warnings A study was conducted to find the effe...

Question

Cigarette Pack Warnings A study was conducted to find the effects of cigarette pack warnings that consisted of text or pictures. Among 1078 smokers given cigarette packs with text warnings, 366 tried to quit smoking. Among 1071 smokers given cigarette packs with warning pictures, 428 tried to quit smoking. (Results are based on data from “Effect of Pictorial Cigarette Pack Warnings on Changes in Smoking Behavior,” by Brewer et al., Journal of the American Medical Association.) Use a 0.01 significance level to test the claim that the proportion of smokers who tried to quit in the text warning group is less than the proportion in the picture warning group.

a. Test the claim using a hypothesis test.

a. Test the claim using a hypothesis test.

Accepted Answer

All right, hello, everyone. So this question says researchers are studying the impact of two different public health campaigns on flu vaccination rates. In the first city, 245 out of 800 residents who saw Campaign X got vaccinated. In the second city, 312 out of 850 residents who saw campaign Y got vaccinated. At the 0.01 significance level, test the claim using a hypothesis test that the proportion of vaccinated residents is lower in the city with campaign X than in the city with campaign Y. So first, let's clarify some terms. Right. For this example, let's say that P1 is the proportion of vaccinated residents for campaign X, and P2 is the proportion of vaccinated residents for campaign Y. So here, the no hypothesis. Is going to state That the proportion of vaccinated residents in campaign X is equal to. The proportion in campaign Y. So P1 Is equal to P2. Now, the alternative hypothesis, on the other hand, states that the proportion of vaccinated residents in campaign X is less than campaign Y. So P1 Would be less than P2. Now let's calculate the proportions themselves, right? We have P1 Equal to the 245 vaccinated residents divided by the total of 800. And this gives you 0.30625. P2 Is equal to 312 divided by 850. And this equals 0.36706. Now let's calculate the pooled proportion. So that's the total vaccinated across both campaigns, divided by the total sample size of both campaigns. So Phat, which represents the total here. Would be the sum of 245 and 312. Divided by the sum of 800 and 850. And this equals approximately 0.33758. And so with this information in mind, we can proceed to calculating the standard error. So the standard error of the differences in populations is as follows. SE Is equal to Phat. Multiplied by one subtracted by Phat. And multiplied by one divided by N1. Added to 1 divided by N 2. Now keep in mind that N1 corresponds to the sample size for campaign X, and N2 corresponds to the sample size for campaign Y. So taking that information into consideration, SE is equal to 0.33758. Multiplied by 1 subtracted by 0.33. 758 Multiplied by And so that's multiplied by 1 divided by 800. Added to 1 divided by 850. And so evaluating this expression gives you a standard error equal to approximately 0.02329. Which leads us now into our test statistic. So here we're computing the Z score. Scroll down once more. The Z score The Z score would be equal to P2 subtracted by P1, and divided by the standard error that we just found. Or actually I had that backwards. It would be P1 subtracted by P2 divided by standard error. So plugging in that information, you would get 0 point. 30625 subtracted by 0.36706. Divided by 0.02329. And this equals approximately -2.61. So now that we have the test statistic, we would determine the critical value and compare the two. Right. So, here, alpha for this experiment is 0.01. So the critical value The critical Z value for a left tailed test at alpha equals 0.01 is approximately -2.33. So here Our calculated Z value -2.61, is less than the critical value of -2.33. Which means that we reject the null hypothesis. So just to clarify, we reject the null hypothesis, which leads us to our final answer, which I went ahead and pre-wrote here on the screen. It reads as follows. Since Z equals -2.61 is less than -2.33, we reject the null hypothesis. Thus we have sufficient evidence to conclude that the proportion of vaccinated residents is significantly lower. In the city with campaign X, then in the city with campaign Y at the 0.01 significance level. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Hypothesis Testing

Significance Level

Proportions and Sample Size

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