Question

10.6.4 Part 1 

A random sample of 500 adult residents of Maricopa County found that 372 were in favor of increasing the highway speed limit to 75 mph, while another sample of 400 adult residents of Pima County found that 288 were in favor of the increased speed limit. Do these data indicate that there is a difference in the support for in increasing the speed limit between the residents of the two counties? Use α = 0.05. 

(a) Test the hypothesis H₀: P1 = P2 versus H1: P1 ≠ P2. What is z₀? Round your answer to two decimal places (e.g. 98.76). 

(b) Is it reasonable to conclude that there is a difference in the support for increasing the speed limit between the residents of the two counties? 

10.6.4 Part 2 A random sample of 500 adult residents of Maricopa County found that 364 were in favor of increasing the highway speed limit to 75 mph, while another sample of 400 adult residents of Pima County found that 279 were in favor of the increased speed limit. Construct a 95% confidence interval on the difference in the two proportions. Round your answer to four decimal places (e.g. 98.7654). 

Answer 1

1) a) $\widehat p1$ = 372/500 = 0.744

$\widehat p2$ = 288/400 = 0.72

The pooled sample proportion (P) = ($\widehat p1$ * n1 + $\widehat p2$ * n2)/(n1 + n2) = (0.744 * 500 + 0.72 * 400)/(500 + 400) = 0.7333

SE = sqrt(P(1 - P)(1/n1 + 1/n2))

= sqrt(0.7333 * (1 - 0.7333) * (1/500 + 1/400))

= 0.0297

The test statistic is

$z_{0} = \frac{\widehat p1 - \widehat p2}{SE}$

$= \frac{0.744 - 0.72}{0.0297}$

  

b) P-value = 2 * P(Z > 0.81)

= 2 * (1 - P(Z < 0.81))

= 2 * (1 - 0.7910)

= 0.4180

Since the P-value is greater than the significance level (0.4180 > 0.05), so we should not reject the null hypothesis.

No, it is not reasonable to conclude that there is a difference in the support for increasing the speed limit between the residents of the two countries.

2) $\widehat p1$ = 364/500 = 0.728

$\widehat p2$ = 279/400 = 0.6975

The pooled sample proportion (P) = ($\widehat p1$ * n1 + $\widehat p2$ * n2)/(n1 + n2) = (0.728 * 500 + 0.6975 * 400)/(500 + 400) = 0.7144

SE = sqrt(P(1 - P)(1/n1 + 1/n2))

= sqrt(0.7144 * (1 - 0.7144) * (1/500 + 1/400))

= 0.0303

At 95% confidence level, the critical value is z0.025 = 1.96

The 95% confidence interval is

$(\widehat p1 - \widehat p2) \pm z_{0.025} * SE$

= (0.728 - 0.6975) +/- 1.96 * 0.0303

= 0.0305 +/- 0.0594

= -0.0289, 0.0899