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Chapter 10: Problem 44

The 2018 Pew Research poll in exercise \(10.43\) also reported responses by political party. Survey results found \(45 \%\) of Republicans and \(69 \%\) of Democrats 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 political party for these two groups using a significance level of \(0.05 .\) c. Does this suggest that these political parties differ significantly in their support of marijuana legalization?

Short Answer

Expert verified
The answer depends on the results of the Chi-Square Test in Step 3 and the decision made in Step 5. The null hypothesis will either be rejected or not rejected based on the p-value. This result will suggest whether the political parties differ significantly in their support of marijuana legalization.

Step by step solution

01

Fill in the Two-Way Table

Start by calculating the count of Republicans and Democrats who support and do not support marijuana legalization. Based on the percentages provided, you can calculate the numbers by multiplying the respective percentages by 200 (the sample size).For Republicans, \( 200 \times 0.45 = 90 \) are in support; hence \( 200 - 90 = 110 \) do not support. For Democrats, \( 200 \times 0.69 = 138 \) are in support; hence \( 200 - 138 = 62 \) do not support.
02

Form null and alternative hypotheses

Null hypothesis \( H_0 \): Support of marijuana legalization is independent of political party. Alternative hypothesis \( H_a \): Support of marijuana legalization is not independent of political party.
03

Perform the Chi-Square Test

Now calculate the expected counts using the formula: \[ \text{Expected count} = \frac{(\text{Row total} \times \text{Column total})}{\text{Grand total}} \] Use these expected counts to calculate Chi-Square statistic: \[ \chi^2 = \sum \frac{(O-E)^2}{E} \] where \( O \) represent observed count and \( E \) stands for expected count.
04

Find the p-value

Determine the degrees of freedom \((df)\). For a 2x2 table, \(df = (2-1) \times (2-1) = 1\). Use the Chi-Square distribution table, or a calculator with chi-square capabilities, to find the p-value associated with the calculated chi-square value and the degrees of freedom.
05

Make a decision

Compare the p-value with the given significance level (0.05). If the p-value 鈮� 0.05, reject the null hypothesis. If the p-value > 0.05, fail to reject the null hypothesis.
06

Interpret the results

If you reject the null hypothesis, it indicates that the alternative hypothesis is true, meaning support for marijuana is not independent of party affiliation. If the null hypothesis is not rejected, it means that the data does not provide strong evidence against the null hypothesis, suggesting that support for marijuana may be independent of the party.

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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, also known as a contingency table, is used to summarize the relationship between two categorical variables. In our exercise, the table displays data about marijuana legalization support with categories being political affiliation (Republican or Democrat) and response (support or not support).
  • Rows and Columns: Each row represents a political party, while each column represents their stance on the issue.
  • Cell Entries: These are the counts derived from the percentages given in the exercise. For example, to find how many Republicans support legalization, multiply the percentage (45%) by the sample size (200), resulting in 90.
This table helps visualize and organize the data efficiently, making it easier to perform further statistical analysis.
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. In this exercise, it's used to test if support for marijuana legalization is independent of political party affiliation.
  • Null Hypothesis ( $H_0$ ): This suggests no relationship exists, meaning support is independent of party.
  • Alternative Hypothesis ( $H_a$ ): This proposes a relationship, meaning support is not independent.
The goal is to determine whether the observed data provides enough evidence to reject the null hypothesis. This is typically done using a statistical test, like the Chi-Square test, which analyzes how expected counts compare to observed counts in a two-way table.
Significance Level
The significance level, often denoted as \(\alpha\), is the threshold used to determine whether a hypothesis test's results are statistically meaningful. In this exercise, the significance level is set at 0.05.
  • Choosing \(\alpha\): This value represents a trade-off between Type I (false positives) and Type II errors (false negatives). A level of 0.05 means you are willing to accept a 5% chance of rejecting the null hypothesis when it's actually true.
  • Decision Making: After performing the test, the calculated p-value is compared with the significance level. If the p-value 鈮� 0.05, the null hypothesis is rejected. This suggests a statistical significance indicating a relationship between party affiliation and support.
This threshold helps determine the strength of evidence against the null hypothesis, guiding your conclusion based on the sample data analysis.

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Most popular questions from this chapter

A 2018 Gallup poll asked college graduates if they agreed that the courses they took in college were relevant to their work and daily lives. The respondents were also classified by their field of study. If we wanted to test whether there was an association between response to the question and the field of study of the respondent, should we do a test of independence or homogeneity?

Suppose a polling organization asks a random sample of people if they are Democrat, Republican, or Other and asks them if they think the country is headed in the right direction or the wrong direction. If we wanted to test whether party affiliation and answer to the question were associated, would this be a test of homogeneity or a test of independence? Explain.

In the study described in \(10.35\) researchers also asked survey respondents if they had heard of the HPV vaccine. Data are shown in the table. Test the hypothesis that knowledge of the vaccine and race are associated. Use a \(0.05\) significance level. $$ \begin{array}{|lcc|} \hline \text { Heard of HPV vaccine } & \text { AAPI } & \text { White } \\ \hline \text { Yes } & 248 & 1737 \\ \hline \text { No } & 103 & 193 \\ \hline \end{array} $$

A penny was spun on a hard, flat surface 50 times, and the result was 15 heads and 35 tails. Using a chisquare test for goodness of fit, test the hypothesis that the coin is biased, using a \(0.05\) level of significance.

Professional musicians listened to five violins being played, without seeing the instruments. One violin was a Stradivarius, and the other four were modern-day violins. When asked to pick the Stradivarius (after listening to all five), 39 got it right and 113 got it wrong. a. Use the chi-square goodness-of-fit test to test the hypothesis that the experts are not simply guessing. Use a significance level of \(0.05\). b. Perform a one-proportion \(z\) -test with the same data, using a one-tailed alternative that the experts should get more than \(20 \%\) correct. Use a significance level of \(0.05\). c. Compare your p-values and conclusions.
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