Question

According to the 14th Annual RBC Homeownership Survey conducted by Ipsos Reid in 2007, most Canadians thought purchasing a home is a good investment. Additionally, there was less concern about interest and mortgage rate hikes than at the same time the year before: 51% were concerned about interest rate increases in 2007 versus 56% in 2006; 43% thought mortgage rates would go up in 2007 versus 70% in 2006. Suppose that these results are based on 1000 randomly selected adult Canadians.

(a) (3 pts) Construct a 93% confidence interval for the proportion of Canadians who were concerned about the mortgage rate increase in 2007.

(b) (3 pts) Construct a 98% confidence interval for the difference in the proportion of Canadians who were concerned about the mortgage rate increase in two years.

(c) (7 pts) Use the p-value method to test if the proportion of Canadians who were concerned about the interest rate has decreased from 2006 to 2007 at the significant level 1%. (d) (3 pts) Interpret the p-value for the test in part (c).

Answer 1

(a)

n = 1000    

p = 0.43    

% = 93    

Standard Error, SE = √{p(1 - p)/n} =    √(0.43(1 - 0.43))/1000 = 0.01565567

z- score = 1.811910673    

Width of the confidence interval = z * SE =     1.8119106729526 * 0.0156556698994326 = 0.02836668

Lower Limit of the confidence interval = P - width =     0.43 - 0.0283666753830047 = 0.40163332

Upper Limit of the confidence interval = P + width =     0.43 + 0.0283666753830047 = 0.45836668

The 93% confidence interval is [0.4016, 0.4584]

(b)

n1 = 1000

n2 = 1000

p1 = 0.7

p2 = 0.43

% = 98

Pooled Proportion, p = (n1 p1 + n2 p2)/(n1 + n2) = (1000 * 0.7 + 1000 * 0.43)/(1000 + 1000) = 0.565

q = 1 - p = 1 - 0.565 = 0.435

SE = √(pq * ((1/n1) + (1/n2))) = √(0.565 * 0.435 * ((1/1000) + (1/1000))) = 0.022170927

z- score = 2.326347874

Width of the confidence interval = z * SE = 2.32634787404085 * 0.0221709269089048 = 0.051577289

Lower Limit of the confidence interval = (p1 - p2) - width = 0.27 - 0.0515772886800458 = 0.218422711

Upper Limit of the confidence interval = (p1 - p2) + width = 0.27 + 0.0515772886800458 = 0.321577289

The 98% confidence interval is [0.2184, 0.3216]

(c)

Data:

n1 = 1000

n2 = 1000

p1 = 0.51

p2 = 0.56

Hypotheses:

Ho: p1 ≥ p2

Ha: p1 < p2

Decision Rule:

α = 0.01

Reject Ho if the test p- value < 0.01

Test Statistic:

Average proportion, p = (n1p1 + n2p2)/(n1 + n2) = (1000 * 0.51 + 1000 * 0.56)/(1000 + 1000) = 0.535

q = 1 - p = 1 - 0.535 = 0.465

SE = √[pq * {(1/n1) + (1/n2)}] = √(0.535 * 0.465 * ((1/1000) + (1/1000))) = 0.022305829

z = (p1 - p2)/SE = (0.51 - 0.56)/0.0223058288346342 = -2.24156656

p- value = 0.0124947

Decision:

Since 0.01249 > 0.01, we fail to reject Ho

Conclusion:

There is no sufficient evidence that the interest rate has decreased.