Question

Let π₁ represent the population proportion of U.S. Senate and Congress (House of Representatives) Democrats who are in favor of a new modest tax on "junk food". Let π₂ represent the population proportion of U.S. Senate and Congress (House of Representative) Republicans who are in favor of a new modest tax on "junk food". Out of the 275 Democratic senators and congressmen 82 of them are in favor of a "junk food" tax. Out of the 285 Republican senators and congressmen 60 of them are in favor a "junk food" tax. Using a significance level of α = .01, can we conclude that the proportion of Democrats who favor a “junk food” tax is more than the proportion of Republicans who favor the new tax?

Note: Do not do any intermediate rounding in your calculations!

H₀: (Click to select)μ1p1s1n1π1x-bar1σ1 (Click to select)=≠≤>≥< (Click to select)μ2s2x-bar2π2n2p2σ2

H_A: (Click to select)μ1x-bar1π1p1σ1s1n1 (Click to select)=≠≤>≥< (Click to select)μ2x-bar2π2p2σ2s2n2

Using only the appropriate statistical table in your textbook, the critical value for rejecting H₀ is (Click to select)+-± . (report your answer to 2 decimal places, using conventional rounding rules)

Using the sample data, the calculated value of the test statistic is (Click to select)+-± . (report your answer to 2 decimal places, using conventional rounding rules)

Should the null hypothesis be rejected? (Click to select)yesno

Can you conclude that the proportion of Democrats who favor a “junk food” tax is more than the proportion of Republicans who favor the new tax? (Click to select)yes or no

Using only the appropriate statistical table in your textbook, what is the p-value of this hypothesis test?      

Answer: p-value = _______ (report your answer to 4 decimal places, using conventional rounding rules)

Answer 1

Answer 2

To test whether the proportion of Democrats who favor a "junk food" tax is more than the proportion of Republicans who favor the new tax, we can conduct a two-sample proportion hypothesis test.

Null Hypothesis (H0): The proportion of Democrats who favor a "junk food" tax (π1) is equal to or less than the proportion of Republicans who favor the new tax (π2).

Alternative Hypothesis (HA): The proportion of Democrats who favor a "junk food" tax (π1) is greater than the proportion of Republicans who favor the new tax (π2).

Let's proceed with the hypothesis test:

Step 1: Calculate the sample proportions: Sample proportion for Democrats (p1) = 82/275 = 0.2982 Sample proportion for Republicans (p2) = 60/285 = 0.2105

Step 2: Calculate the pooled sample proportion (p): p = (x1 + x2) / (n1 + n2) p = (82 + 60) / (275 + 285) p = 0.2521

Step 3: Calculate the standard error (SE) of the difference in proportions: SE = sqrt( p * (1 - p) * ((1/n1) + (1/n2)) ) SE = sqrt( 0.2521 * (1 - 0.2521) * ((1/275) + (1/285)) ) SE ≈ 0.0385

Step 4: Calculate the test statistic (Z): Z = (p1 - p2) / SE Z = (0.2982 - 0.2105) / 0.0385 Z ≈ 2.28

Step 5: Find the critical value and p-value: Since the alternative hypothesis is one-tailed (π1 > π2), we will find the critical value from the Z-table corresponding to a significance level of α = 0.01. The critical value is approximately 2.33.

The p-value for the test statistic Z = 2.28 can be found using the Z-table or a statistical calculator. The p-value is the probability of getting a Z-value as extreme as 2.28 or greater, and it is the probability of observing such a difference between proportions under the assumption that the null hypothesis is true.

Step 6: Compare the test statistic and critical value: The test statistic (Z = 2.28) is less than the critical value (2.33). Thus, we do not have enough evidence to reject the null hypothesis.

Step 7: Compare the p-value and significance level (α): The p-value is the probability of observing the data under the assumption of the null hypothesis. If the p-value is less than or equal to the significance level (α = 0.01), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the p-value is greater than 0.01. Thus, we fail to reject the null hypothesis.

Conclusion: Based on the given data and a significance level of α = 0.01, we do not have sufficient evidence to conclude that the proportion of Democrats who favor a "junk food" tax is more than the proportion of Republicans who favor the new tax.

Answer 3

To test whether the proportion of Democrats who favor a "junk food" tax is more than the proportion of Republicans who favor the new tax, we will perform a two-sample proportion hypothesis test.

Hypotheses: H0: π1 ≤ π2 (The proportion of Democrats who favor the tax is less than or equal to the proportion of Republicans who favor the tax) HA: π1 > π2 (The proportion of Democrats who favor the tax is greater than the proportion of Republicans who favor the tax)

Given data: n1 = 275 (number of Democratic senators and congressmen) x1 = 82 (number of Democrats in favor of the tax) n2 = 285 (number of Republican senators and congressmen) x2 = 60 (number of Republicans in favor of the tax) α = 0.01 (significance level)

Step 1: Calculate the sample proportions: p̂1 = x1 / n1 = 82 / 275 ≈ 0.2982 p̂2 = x2 / n2 = 60 / 285 ≈ 0.2105

Step 2: Calculate the standard error: SE = sqrt((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2)) SE ≈ sqrt((0.2982 * 0.7018 / 275) + (0.2105 * 0.7895 / 285)) ≈ 0.0394

Step 3: Calculate the test statistic: Z = (p̂1 - p̂2) / SE Z ≈ (0.2982 - 0.2105) / 0.0394 ≈ 2.2259

Step 4: Find the critical value: Since it is a one-tailed test, and the significance level is 0.01, the critical value can be found from the z-table or a statistical calculator. The critical value for a 0.01 significance level is approximately 2.33.

Step 5: Compare the test statistic with the critical value: Z (2.2259) < Critical value (2.33)

Step 6: Determine the p-value: The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. Using the Z-table, the p-value corresponding to a test statistic of 2.2259 is approximately 0.0131.

Step 7: Conclusion: Since the test statistic (Z) is less than the critical value and the p-value (0.0131) is less than the significance level (0.01), we reject the null hypothesis (H0). There is evidence to suggest that the proportion of Democrats who favor a "junk food" tax is more than the proportion of Republicans who favor the new tax.

Conclusion: Yes, we can conclude that the proportion of Democrats who favor a "junk food" tax is more than the proportion of Republicans who favor the new tax.