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Answer :
Final answer:
This is a hypothesis testing problem for comparing two proportions. The null hypothesis states there is no difference between the proportions of Rhode Island and Massachusetts college-age residents supporting the legalization of recreational marijuana. If the calculated p-value is lower than the significance level 0.05, we reject the null hypothesis and conclude there's a significant difference, with Massachusetts having a higher proportion of supporters.
Explanation:
The objective here is to test if the proportion of college-age Massachusetts residents who support the legalization of recreational marijuana exceeds the proportion of college-age Rhode Island residents who support it. This is a hypothesis test for comparing two proportions.
Here we define the null hypothesis (H0) as the proportion of supporters in Rhode Island (p1) is equal to or greater than the proportion of supporters in Massachusetts (p2). The alternative hypothesis (H1) would then be defined as p1 < p2.
We'd use the sample data to calculate the pooled sample proportion (p) and standard error (SE). Then, compute the z-score for the observed sample difference (p1-p2), from where we'd get the observed significance level (p-value) to compare with the provided significance level (0.05).
If the p-value is less than 0.05, then we would reject the null hypothesis, concluding that there is a significant difference in the proportion of supporters of the legalization of recreational marijuana between college-age residents of Massachusetts and Rhode Island, and the proportion is higher for Massachusetts.
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Perform a two-sample z-test for proportions to test if the proportion of college-age Massachusetts residents who support the legalization of recreational marijuana exceeds the proportion of college-age Rhode Island residents who support the legalization of recreational marijuana.
We are comparing two independent proportions, so we need to perform a two-sample z-test for proportions.
Let p1 be the proportion of college-age Rhode Island residents who support the legalization of recreational marijuana, and p2 be the proportion of college-age Massachusetts residents who support the legalization of recreational marijuana.
The null hypothesis is that the proportion of college-age Massachusetts residents who support the legalization is less than or equal to the proportion of college-age Rhode Island residents who support the legalization:
H0: p2 <= p1
The alternative hypothesis is that the proportion of college-age Massachusetts residents who support the legalization is greater than the proportion of college-age Rhode Island residents who support the legalization:
Ha: p2 > p1
We can use the pooled proportion to estimate the common proportion for both groups:
p = (x1 + x2) / (n1 + n2)
where x1 and x2 are the number of supporters in each group, and n1 and n2 are the sample sizes.
The test statistic is calculated as:
z = (p2 - p1) / sqrt(p*(1-p)*(1/n1 + 1/n2))
Under the null hypothesis, the test statistic follows a standard normal distribution.
Using the given values, we have:
p1 = 183/300 = 0.61
n1 = 300
p2 = 238/350 = 0.68
n2 = 350
p = (183 + 238) / (300 + 350) = 0.644
Plugging in these values, we get:
z = (0.68 - 0.61) / sqrt(0.644*(1-0.644)*(1/300 + 1/350)) = 2.33
Using a standard normal distribution table, the p-value is approximately 0.01.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that the proportion of college-age Massachusetts residents who support the legalization of recreational marijuana is significantly greater than the proportion of college-age Rhode Island residents who support the legalization.
Therefore, we have evidence to suggest that the legalization of recreational marijuana is more supported among college-age Massachusetts residents compared to college-age Rhode Island residents.
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