Suppose a researcher wants to investigate whether the...

Suppose a researcher wants to investigate whether the proportion of smokers within different age groups is the same. He divides the American population into four age groups: 18 to 29 years old, 30 to 49 years old, 50 to 64 years old, and 65 years or older. Within each age group, he surveys 80 individuals and asks, "Have you smoked at least one cigarette in the past week?" The results of the survey are as follows: (a) Is there evidence to indicate that the proportion of individuals within each age group who have smoked at least one cigarette in the past week is different at the $\alpha=0.05$ level of significance? (b) Compute the $P$ -value for this test by finding the area under the chi-square distribution to the right of the test statistic. (c) Construct a conditional distribution by age and draw a bar graph. Does this evidence support your conclusion in part (a)?

Suppose a researcher wants to investigate whether the proportion of smokers within different age groups is the same. He divides the American population into four age groups: 18 to 29 years old, 30 to 49 years old, 50 to 64 years old, and 65 years or older. Within each age group, he surveys 80 individuals and asks, "Have you smoked at least one cigarette in the past week?" The results of the survey are as follows:
(a) Is there evidence to indicate that the proportion of individuals within each age group who have smoked at least one cigarette in the past week is different at the $\alpha=0.05$ level of significance?
(b) Compute the $P$ -value for this test by finding the area under the chi-square distribution to the right of the test statistic.
(c) Construct a conditional distribution by age and draw a bar graph. Does this evidence support your conclusion in part (a)?

Key Concepts

Chi-Square Test for Homogeneity

This test examines whether different populations or groups have the same distribution of a categorical variable. It compares the observed frequencies in each category across several groups to the frequencies that would be expected if the populations were homogeneous, providing a method to assess differences between groups in terms of proportions or percentages.

Hypothesis Testing Framework

In the context of categorical data, hypothesis testing involves formulating a null hypothesis that assumes no difference in the population proportions, and an alternative hypothesis that indicates at least one group differs. The test then uses a test statistic alongside a chosen significance level (alpha) to decide whether to reject the null hypothesis, allowing researchers to draw conclusions based on the data.

P-value Interpretation

The P-value in hypothesis testing represents the probability of observing test results at least as extreme as those measured, under the assumption that the null hypothesis is true. In chi-square tests, the area in the right tail of the chi-square distribution beyond the computed test statistic is used to determine this likelihood, guiding the decision to accept or reject the null hypothesis.

Conditional Distribution in Contingency Tables

A conditional distribution in a contingency table provides the proportions or percentages of one categorical variable within the levels of another categorical variable. Constructing these distributions helps in understanding the relationship between variables by allowing comparisons across different groups, thus complementing the formal hypothesis test.

Data Visualization for Categorical Data

Visual tools such as bar graphs are used to represent and explore categorical data distributions clearly and succinctly. They present the proportions or counts of different categories and can aid in quickly identifying patterns or differences between groups, thereby supporting conclusions drawn from statistical tests.

Transcript

00:01 So we're going to be assuming that for each of the smoking versus non -smoking, that the proportion in age category 1 for smokers are the same.

00:15 So for category 1, the proportion of smokers is equal to the non -smokers.

00:22 For category 2, that the proportion of smokers versus non -smokers is the same, et cetera, for all four.

00:31 So this is true for all four age categories.

00:42 And alternately, they're not all proportions for each of them are equal.

00:52 And the kye squared statistic we get in our number of degrees of freedom will end up being the number of rows is two, so less one.

01:00 The number of columns is four less one, so we have three degrees of freedom and that statistic ironically comes out to be six and if we're using a 5 % significance level and we draw a little picture of a distribution we put all 0 .5 here for three degrees of freedom that cutoff point or that critical value is 7 .81 and we have a value of six so we would fail to reject null to the the null.

01:35 So it would appear as though these proportions are approximately equal for each of the age categories.

01:41 Part b we wanted to find what the p value is.

01:44 And that p value is this area.

01:47 And we know that area is going to be greater than 0 .05.

01:50 Since we did not, we failed to reject the null.

01:53 And that comes out to be 0 .1116.

01:57 That's our p value.

01:59 Now if we wanted to graph a conditional distribution, and i'm going to let green be a smoker and i'll let red stand for a non -smoker and if i quick look at my my conditional distributions it looks like the highest one is about about 28 percent so if i have this be there's 10 .1 .2 .0 .3 there's 0 .4...

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