Compute the test statistic used to test the null hypothesis that p 1 = p 2
-In a vote on the Clean Water bill, 45% of the 205 Democrats voted for the bill while 52% of the 230 Republicans voted for it.
A)-1.604
B)-1.239
C)-0.875
D)-1.458

Question

Compute the test statistic used to test the null hypothesis that p 1 = p 2 
-In a vote on the Clean Water bill, 45% of the 205 Democrats voted for the bill while 52% of the 230 Republicans voted for it.
A)-1.604
B)-1.239
C)-0.875
D)-1.458

Quizplus · Accepted Answer

The test statistic for comparing two proportions is calculated using the formula: $$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{P(1-P)(\frac{1}{n_1} + \frac{1}{n_2})}}$$ where $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions, $n_1$ and $n_2$ are the sample sizes, and $P$ is the pooled proportion, calculated as $$P = \frac{x_1 + x_2}{n_1 + n_2}$$ where $x_1$ and $x_2$ are the number of successes in each sample.Given:- 45% of 205 Democrats voted for the bill, so $\hat{p}_1 = 0.45$ and $n_1 = 205$.- 52% of 230 Republicans voted for it, so $\hat{p}_2 = 0.52$ and $n_2 = 230$.First, calculate the number of successes in each group:- Democrats: $x_1 = 0.45 \times 205 = 92.25$, but since the number of people must be whole, consider $x_1 = 92$.- Republicans: $x_2 = 0.52 \times 230 = 119.6$, similarly consider $x_2 = 120$.Then, calculate the pooled proportion $P$:$$P = \frac{92 + 120}{205 + 230} = \frac{212}{435}$$Finally, calculate the test statistic $z$:$$z = \frac{0.45 - 0.52}{\sqrt{P(1-P)(\frac{1}{205} + \frac{1}{230})}}$$Plugging in the values and calculating gives a test statistic of approximately -1.458, which matches choice D.

Compute the Test Statistic Used to Test the Null Hypothesis