Incentives for Quitting Smoking: Group or Individual? In a...

Incentives for Quitting Smoking: Group or Individual? In a smoking cessation program, over 2000 smokers who were trying to quit were randomly assigned to either a group program or an individual program. After six months in the program, 148 of the 1080 in the group program were successfully abstaining from smoking. while 120 of the 990 in the individual program were successful. ${ }^{15}$ We wish to test to see if this data provide evidence of a difference in the proportion able to quit smoking between smokers in a group program and smokers in an individual program. (a) State the null and alternative hypotheses, and give the notation and value of the sample statistic. (b) Use a randomization distribution and the observed sample statistic to find the p-value. (c) Give the mean and standard error of the normal distribution that most closely matches the randomization distribution, and then use this normal distribution with the observed sample statistic to find the p-value. (d) Use the standard error found from the randomization distribution in part (b) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution. (e) Compare the p-values from parts (b), (c), and (d). Use any of these p-values to give the conclusion of the test.

Incentives for Quitting Smoking: Group or Individual? In a smoking cessation program, over 2000 smokers who were trying to quit were randomly assigned to either a group program or an individual program. After six months in the program, 148 of the 1080 in the group program were successfully abstaining from smoking. while 120 of the 990 in the individual program were successful. ${ }^{15}$ We wish to test to see if this data provide evidence of a difference in the proportion able to quit smoking between smokers in a group program and smokers in an individual program.
(a) State the null and alternative hypotheses, and give the notation and value of the sample statistic.
(b) Use a randomization distribution and the observed sample statistic to find the p-value.
(c) Give the mean and standard error of the normal distribution that most closely matches the randomization distribution, and then use this normal distribution with the observed sample statistic to find the p-value.
(d) Use the standard error found from the randomization distribution in part (b) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution.
(e) Compare the p-values from parts (b), (c), and (d). Use any of these p-values to give the conclusion of the test.

Key Concepts

Standardized Test Statistic

The standardized test statistic (often a z-score when using the normal approximation) measures how many standard errors the observed sample statistic is away from the hypothesized value (typically zero difference under the null hypothesis). This standardization allows the use of statistical tables or software to determine the p-value, facilitating the decision on whether to reject the null hypothesis. It is a bridge between the observed data and the theoretical distribution used to assess significance.

Standard Error

The standard error quantifies the variability of the sample statistic (such as the difference in proportions) from sample to sample. It is an essential component in forming the test statistic, as it provides a measure of the precision of the estimate. In the context of comparing two proportions, the standard error is calculated based on the proportions and sample sizes, and it is used both in the construction of the randomization distribution and the normal approximation.

Normal Approximation

For large samples, the central limit theorem allows the distribution of the difference in sample proportions to be approximated by a normal distribution. This approximation simplifies the process of hypothesis testing by enabling the use of z-scores to compute p-values and confidence intervals, even though the underlying data might be binomial. The normal approximation is particularly useful for facilitating analytical solutions in tests comparing proportions.

p-value

The p-value represents the probability of observing a test statistic at least as extreme as the one calculated from the data, under the assumption that the null hypothesis is true. It provides a quantitative measure to assess the strength of the evidence against the null hypothesis. A small p-value indicates that the observed data are unlikely to have occurred by chance alone, suggesting that the alternative hypothesis may be true. It is a central concept in hypothesis testing and decision making.

Difference in Proportions

When comparing two independent groups, an important key concept is the difference in their proportions (success rates, for example). The sample statistic calculated as the difference between the two sample proportions serves as an estimator for the true difference in the population proportions. This concept is central to understanding how the observed data relate to the hypothesis being tested.

Hypothesis Testing

This concept involves formulating a null hypothesis (typically stating that there is no difference or effect) and an alternative hypothesis (indicating the presence of a difference or effect). In the context of comparing two proportions, the null hypothesis usually posits that the proportions are equal, while the alternative hypothesis suggests that they are not. This fundamental step sets the stage for inference and aids in determining whether the observed data provide enough evidence to reject the null hypothesis in favor of the alternative.

Randomization Distribution

A randomization (or permutation) distribution is an empirical distribution of the test statistic generated by recalculating it repeatedly under rearranged assignments of the observed outcomes. It reflects the sampling variability under the null hypothesis and is particularly useful when the assumptions for standard parametric methods (like normality) may not hold or when an exact test is desired. This distribution is used to determine the likelihood of the observed difference occurring by chance.

Transcript

00:01 This problem, we have 2 ,000 smokers who are trying to quit, and they're randomly assigned to either a group program or an individualized program.

00:22 And after six months in the program, there were 1 ,080 in the group program, and 148 of them were successful.

00:35 And there were 990 in the individualized program and 200 or 120 of them were successful.

00:49 And we wish to see if the data provides evidence of a difference in the proportions.

00:56 So we've got to look at each proportion.

00:58 So our point estimate for the group proportion of successful individuals was 148 out of 1 .4.

01:08 And the point estimate for the individualized success was 120 out of 990.

01:21 And again, what we are trying to test is to see if there is a difference between group cessation programs and individualized.

01:34 So in part a, we want to generate our hypotheses, our null hypothesis, is going to be that there is no difference between group cessation programs and individualized, and our alternative will be that there is a difference.

02:05 We also want to calculate our sample statistic, and our sample statistic is going to be the difference.

02:23 Between the two.

02:30 So we're taking 148 out of 180 minus 120 over 990.

02:40 As a decimal, that will get you a value of 0 .0158, 249 -158.

02:54 So we're going to round that to three decimal places, 016 .016.

03:02 Now for part b, we want to use a randomized distribution to help us calculate the standard error and the p value.

03:36 In order to do this, we're going to use the app stat key.

03:39 So i'm going to bring in our stat key app.

03:42 And across the top, we are going to go to the test for difference in proportions.

03:51 We will have to edit our data.

03:54 So in group 1, 148 were successful out of 1 ,080 in the sample.

04:09 And in group 2, 120 were successful out of 990 in the sample.

04:19 Now, because our null hypothesis is an inequality, it is a two -tailed test.

04:26 We're going to click on the two -tailed test.

04:29 And we are now going to generate 1 ,000 samples.

04:35 Now, in doing so, you can see that we have red areas in the two different tails of this randomized distribution.

04:46 So we want our tails to correspond with our sample statistic.

04:51 So we're going to click on the boundary line, the 0 .27, and we're going to edit that to be in line.

05:01 With our sample statistic, which is .016.

05:18 So now you could see the red areas in the two tails has altered a little bit.

05:25 And we're going to generate a couple more samples.

05:28 Right now we have a thousand samples in our distribution.

05:32 Let's generate 2 ,000, let's generate another thousand, let's generate another thousand, and we'll go one more.

05:45 So we have generated 5 ,000 samples, and you could see that in the upper right hand corner, we have a standard error, and we have our two red areas, and our p value will be the sum of those two red areas.

06:02 So we're going to add 0 .125 plus 0 .125, and we will get 0 .2 .0 .000.

06:13 Now, continue to notice that as i generate more samples, so if i generated another thousand, those areas do change, and i generate more, and those areas continue to change.

06:30 So using the randomization distribution is a good estimate, but it's, again, going to continue to change as our sample size gets larger.

06:39 But as you noticed, our standard error did very little changing.

06:43 So in part c, we want to determine our p value again, but in the meantime, we want to give the mean and standard error of the normal distribution that most closely matches this randomized distribution.

07:02 So we could see it is bell -shaped, so it is approximated by a normal distribution.

07:08 In the center of that bell -shape is a mean of zero, and then we have our standard error.

07:16 Which was 0 .015.

07:19 So our normal distribution is the mean of zero and a standard error of 0 .015.

07:27 So now we want to use that to determine our p value as well.

07:34 So we're going to get out of the randomization section of this app by going back up to the stat key in the upper left corner.

07:42 We're going to click on that...

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