Let p1 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 p2 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". A few years ago, out of the 265 democratic senators and congressman 106 of them were in favor of a "junk food" tax. Out of the 285 republican senators and congressman only 57 of them were in favor a "junk food" tax. Based on this data, at α =.01, can we conclude that the proportion of democrats who favor “junk food" tax is more than 5% higher than proportion of republicans who favor such a tax? Indicate which test you are performing; show the hypotheses, the test statistic and the critical values and mention whether one-tailed or two-tailed.
You will need to use a binomial proportion two-sample z-test using proportions.
Formula:
z = (p1 - p2)/√[pq(1/n1 + 1/n2)]
p1 = 106/265
p2 = 57/285
n1 = 265
n2 = 285
p = (p1 + p2)/(n1 + n2)
q = 1 - p
Convert all fractions to decimals. Plug those decimal values into the formula and find z. Once you have that value, you will be able to compare to the critical or cutoff value you find in the z-table at .01 level of significance for a one-tailed test (hint: you are looking at "greater than" in the alternate hypothesis).
I'll let you take it from here.
To determine whether the proportion of Democrats who favor a "junk food" tax is more than 5% higher than the proportion of Republicans who favor such a tax, you can perform a hypothesis test for the difference between two proportions.
The hypotheses for this test can be stated as follows:
Null Hypothesis (H0): p1 - p2 <= 0 (No significant difference between the proportions)
Alternative Hypothesis (Ha): p1 - p2 > 0 (Proportion of Democrats who favor the tax is more than 5% higher than Republicans)
To calculate the test statistic, you will need to compute the pooled proportion and the standard error.
Pooled proportion (p̂): (x1 + x2) / (n1 + n2)
Where x1 = number of Democrats in favor of the tax (106),
x2 = number of Republicans in favor of the tax (57),
n1 = total number of Democrats (265),
n2 = total number of Republicans (285).
p̂ = (106 + 57) / (265 + 285) = 163 / 550 ≈ 0.2964
Standard Error (SE): sqrt(p̂ * (1 - p̂) * (1/n1 + 1/n2))
SE = sqrt(0.2964 * (1 - 0.2964) * (1/265 + 1/285)) ≈ 0.0349
The test statistic (Z) can be calculated using the following formula:
Z = (p1 - p2 - 0.05) / SE
Where p1 = x1 / n1 and p2 = x2 / n2.
Z = (106 / 265 - 57 / 285 - 0.05) / 0.0349 ≈ 2.071
For a significance level α = 0.01, the critical value for a one-tailed test is Zα = 2.326 (obtained from a standard normal distribution table).
Since the calculated test statistic (Z = 2.071) is less than the critical value (Zα = 2.326), we fail to reject the null hypothesis. Thus, based on the given data, we cannot conclude that the proportion of Democrats who favor a "junk food" tax is more than 5% higher than the proportion of Republicans who favor such a tax.
Note: This test is one-tailed because the alternative hypothesis specifies a directional difference (more than 5% higher for Democrats).