Teen smoking, part I A Vermont study published by the...

Teen smoking, part I A Vermont study published by the American Academy of Pediatrics examined parental influence on teenagers’ decisions to smoke. A group of students who had never smoked were questioned about their parents’ attitudes toward smoking. These students were questioned again two years later to see if they had started smoking. The researchers found that, among the 284 students who indicated that their parents disapproved of kids smoking, 54 had become established smokers. Among the 41 students who initially said their parents were lenient about smoking, 11 became smokers. Do these data provide strong evidence that parental attitude influences teenagers’ decisions about smoking? a) What kind of design did the researchers use? b) Write appropriate hypotheses. c) Are the assumptions and conditions necessary for inference satisfied? d) Test the hypothesis and state your conclusion. e) Explain in this context what your P-value means. f) If that conclusion is actually wrong, which type of error did you commit?

Teen smoking, part I A Vermont study published by the American Academy of Pediatrics examined parental influence on teenagers’ decisions to smoke. A group of students who had never smoked were questioned about their parents’ attitudes toward smoking. These students were questioned again two years later to see if they had started smoking. The researchers found that, among the 284 students who indicated that their parents disapproved of kids smoking, 54 had become established smokers. Among the 41 students who initially said their parents were lenient about smoking, 11 became smokers. Do these data provide strong evidence that parental attitude influences teenagers’ decisions about smoking?
a) What kind of design did the researchers use?
b) Write appropriate hypotheses.
c) Are the assumptions and conditions necessary for inference satisfied?
d) Test the hypothesis and state your conclusion.
e) Explain in this context what your P-value means.
f) If that conclusion is actually wrong, which type of error did you commit?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that involves stating a null hypothesis, which represents no effect or no difference, and an alternative hypothesis, which reflects the effect or difference being investigated. This framework provides a systematic way to determine whether observed data are sufficiently extreme to reject the null hypothesis in favor of the alternative hypothesis.

Types of Errors in Hypothesis Testing

In hypothesis testing, two primary types of errors can occur. A Type I error happens when a true null hypothesis is incorrectly rejected, representing a false positive. Conversely, a Type II error occurs when a false null hypothesis is not rejected, representing a false negative. Understanding these error types is essential for interpreting the implications of the test results and the risks of drawing incorrect conclusions.

P-value Interpretation

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis by quantifying the probability of observing the data, or something more extreme, if the null hypothesis were true. It provides researchers with a metric for deciding whether to reject the null hypothesis, typically by comparing it to a pre-set significance level.

Observational Study Design

This concept refers to research in which the investigator does not manipulate the exposure or treatment but instead observes it in a natural setting. In a prospective cohort study, a group of individuals is selected based on their exposure status and followed over time to assess outcomes. This design is key in understanding relationships and associations, even though it is less conclusive for making causal inferences compared to randomized experiments.

Assumptions for Statistical Inference

Before applying inferential methods, certain underlying assumptions must be verified. These include the independence of observations, representative sampling, appropriate sample sizes, and the proper distribution of data (such as normality in certain tests). Satisfying these conditions ensures that the test results are valid and that inferences about the population are reliable.

Transcript

00:01 So if you read the question, first of all, we're talking about this being a prospective study because they looked at it at the beginning and then they looked at opinions two years later.

00:13 And let's call group a those that the parents initially disapproved of smoking.

00:21 And let's say b is the one, the group that they're more lenient.

00:27 And now we know that p -hat a, that group, that there were a large, number, 284 people, and after two years, 54 of those students had begun smoking.

00:44 And the p hat for b was, there were 41 students, again, a smaller group, of which 11 had started to smoke.

00:55 And so question b asks us, what hypotheses would we use? and our hypotheses we would use is we would assume that the proportion in group a, who end up smoking, is going to equal the proportion in group b.

01:06 And alternately we want to know if there's a difference.

01:11 Just said difference.

01:12 Is there a difference between the two groups? now, we want to have our assumptions.

01:19 Well, we don't know if we have simple random samples.

01:23 We know that the number of people in group a is greater than or equal to 10 times the sample size.

01:30 And we know that the number of people that could potentially be in group b is greater than or equal to 10 times the sample size.

01:37 Size.

01:38 And if we find our pooled p, our pooled p is going to be the 65 out of adding these two together.

01:47 Let's see, that's a five and that adds a 12 carry the one, 325.

01:52 And if we take n times the sample size of that times the pooled p and take that times one minus the pooled p and likewise do the same thing with the sample size of 41 and multiply it by pooled p and the this times one minus the pooled p all of these are going to be greater than are equal to five so as long as if you have a simple random sample the answer is yes our assumptions are going to be met and again these need to be greater than equal to five so now let's perform our test so we're going to be assuming that the difference between these two groups is equal to zero and let's see what do we get the difference to be the difference ends up being let's just get a decimal approximation if i take the 54 divided by 284 minus the 11 divided by 41 we get the difference to be about negative 8 percent so we're getting something down here but again we know we're doing a two -tail test so it ends up being down here at about negative 0 .078 and these two to our p value.

03:11 So let's calculate our test statistic.

03:14 And so what's the likelihood of having a z value that is less than or equal to and we have the two proportions.

03:24 Let me move this here so i can see them.

03:26 54 out of 284 minus 11 of 41...

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