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Chapter 10: Problem 43

A 2018 Pew Research poll asked a random sample of Millennials and GenXers if they supported legalization of marijuana. Survey results found \(70 \%\) of Millennials and \(66 \%\) of GenXers supported marijuana legalization. a. Use these results to fill in the following two-way table with the counts in each category. Assume the sample size for each group was 200 . b. Test the hypothesis that support of marijuana legalization is independent of generation for these two groups using a significance level of \(0.05\). c. Does this suggest that these generations differ significantly in their support of marijuana legalization?

Short Answer

Expert verified
If the chi-squared test statistic is greater than the critical value, there is a significant difference in support for marijuana legalization among Millennials and GenXers. Otherwise, there is no significant difference.

Step by step solution

01

Fill in the Two-Way Table

The given percentages and sample size are used to fill in the following two-way table:\[\begin{array}{ccc}& \text{Support} & \text{Do Not Support} \\text{Millennials} & 0.70 \times 200 & 0.30 \times 200 \\text{GenXers} & 0.66 \times 200 & 0.34 \times 200\end{array}\]which gives us the following filled table:\[\begin{array}{ccc}& \text{Support} & \text{Do Not Support} \\text{Millennials} & 140 & 60 \\text{GenXers} & 132 & 68\end{array}\]
02

Calculate the Chi-Squared Test Statistic

Use the formula for the Chi-Squared Test Statistic, which is:\[\chi ^{2} = \sum \frac{(O - E)^2}{E}\]where O represents observed values and E represents expected values. The expected value can be calculated using the formula:\[E_i = \left( \sum_{\text{row i}} \frac{\sum_{\text{column i}}}{n} \right)\]Applying these formulas onto each cell of the two-way table and summing up all the four results, we get the chi-squared test statistic, which will be compared to the critical value.
03

Determine the Critical Value

To determine the critical value, we will use the Chi-Square distribution table and the Degrees of Freedom (df), which is (number of rows - 1) * (number of columns - 1). For our two-way table, df=(2-1)*(2-1) = 1. Considering the significance level of 0.05, for df = 1, the critical value is approximately 3.841.
04

Make the Decision

If the chi-squared test statistic is greater than the critical value, then we reject the null hypothesis that the support for marijuana legalization is independent of the generation. If it is not greater, then there is not enough evidence to reject the null hypothesis, meaning that the support for marijuana legalization does not depend on the generation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Way Table
A two-way table, sometimes known as a contingency table, helps organize data that involves two variables. In this example, the variables are 'Millennials' and 'GenXers' against their support or opposition to marijuana legalization. The table is a matrix with rows and columns, offering a neat way to compare across variables.
The task begins by observing the given percentages. Millennials with 70% support translates to 140 individuals based on a survey size of 200. Similarly, 66% of GenXers supporting translates to 132 individuals. Complemented with their respective oppositions,
  • Millennials who do not support: 60 (30% of 200)
  • GenXers who do not support: 68 (34% of 200)
This completed two-way table lays a foundation for further statistical testing.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data. In this scenario, we're using it to determine whether support for marijuana legalization is associated with the generation, i.e., if it's independent or not.
Initiating hypothesis testing starts with forming a null hypothesis ( H_0 ) and an alternative hypothesis ( H_a ). Here:
  • H_0 : Support for marijuana legalization is independent of the generation.
  • H_a : Support for marijuana legalization is not independent of generation.
By calculating the Chi-Squared test statistic鈥攃omparing observed counts from the filled two-way table to expected counts, which are derived using row and column totals鈥攖he association or lack thereof can be statistically evaluated.
Significance Level
The significance level, often denoted by 伪 , is a critical aspect of hypothesis testing. It quantifies the risk of wrongly rejecting the null hypothesis. In this task, a significance level of 0.05 is used, reflecting a 5% risk.
After computing the Chi-Squared test statistic, the next step involves comparing it to the critical value from the Chi-Square distribution table. With 1 degree of freedom and a significance level of 0.05, the critical value is approximately 3.841. If the computed statistic exceeds this value, the null hypothesis is rejected. This outcome implies a significant generation-related difference in marijuana legalization support, else the null hypothesis is upheld.
Understanding the role of the significance level aids in making informed, reliable conclusions from hypothesis testing.

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Most popular questions from this chapter

In the study referenced in exercise \(10.33\), researchers also collected data on use of apps to monitor diet and calorie intake. The data are reported in the table. Test the hypothesis that diet app use and gender are associated. Use a \(0.05\) significance level. $$ \begin{array}{ccc} \text { Use } & \text { Male } & \text { Female } \\ \hline \text { Yes } & 43 & 241 \\ \hline \text { No } & 50 & 84 \\ \hline \end{array} $$

In a 2015 study reported in the New England Journal of Medicine, Du Toit et al. randomly assigned infants who were likely to develop a peanut allergy (as measured by having eczema, egg allergies, or both) to either consume or avoid peanuts until 60 months of age. The infants in this cohort did not previously show any preexisting sensitivity to peanut extract. The numbers in each group developing a peanut allergy by 60 months of age are shown in the following table. $$ \begin{array}{lcc} & \text { Treatment Group } \\ \hline \begin{array}{l} \text { Peanut allergy at age } \\ \mathbf{6 0} \text { mos. } \end{array} & \text { Consume peanuts } & \text { Avoid peanuts } \\ \hline \text { Yes } & 5 & 37 \\ \hline \text { No } & 267 & 233 \\ \hline \end{array} $$ a. Compare the percentages in each group that developed a peanut allergy by age 60 months. b. Test the hypothesis that treatment group and peanut allergy are associated using the chi-square statistic. Use a significance level of \(0.05\). c. Do a Fisher's Exact Test for the data with the same significance level. Report the two-tailed p-value and your conclusion. (Use technology to run the test.) d. Compare the p-valucs for parts \(\mathrm{b}\) and \(\mathrm{c}\). Which is morc accurate? Explain.

When playing Dreidel, (see photo) you sit in a circle with friends or relatives and take turns spinning a wobbly top (the dreidel). In the center of the circle is a pot of several foil-wrapped chocolate coins. If the four-sided top lands on the Hebrew letter gimmel, you take the whole pot and everyone needs to contribute to the pot again. If it lands on hey, you take half the pot. If it lands on nun, nothing happens. If it lands on \(\operatorname{shin}\), you put a coin in. Then the next player takes a turn. Each of the four outcomes is believed to be equally likely. One of the author's families got the following outcomes while playing with a wooden dreidel during Hanukah. Determine whether the outcomes allow us to conclude that the dreidel is biased (the four outcomes are not equally likely). Use a significance level of \(0.05\). $$ \begin{array}{cccc} \text { gimmel } & \text { hey } & \text { nun } & \text { shin } \\ 5 & 1 & 7 & 27 \end{array} $$

Breakfast Habits (Example \(1 \&\) 2) In a 2015 study by Nanney et al. and published in the Journal of American College Health. a random sample of community college students was asked whether they ate breakfast 3 or more times weekly. The data are reported by gender in the table. $$ \begin{array}{|lcc|} \hline \text { Eat breakfast at least } 3 \times \text { weekly } & \text { Females } & \text { Males } \\ \hline \text { Yes } & 206 & 94 \\ \hline \text { No } & 92 & 49 \\ \hline \end{array} $$ a. Find the row, column, and grand totals, and prepare a table showing these values as well as the counts given. b. Find the percentage of students overall who eat breakfast at least three times weekly. Round off to one decimal place. c. Find the expected number who eat breakfast at least three times weekly for each gender. Round to two decimal places as needed. d. Find the expected number who did not eat breakfast at least three times weekly for each gender. Round to two decimal places as needed. e. Calculate the observed value of the chi-square statistic.

Treatment In a 2018 study by Zhu et al. reported in The Lancet, researchers conducted an experiment to determine the efficacy and safety of the drug dorzagliatin in the treatment of patients with Type 2 diabetes. In this double-blind study, patients were randomly assigned to one of two treatment groups (drug or placebo) and the glucose levels of the two groups were compared after 12 weeks. If we test whether treatment group is associated with glucose level, are we doing a test of homogeneity or a test of independence?
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