Question

A hypothetical investigation on rider satisfaction with a particular public transit system serving commuting residents of British California (BC) and Prince Edward’s County (PEC) offers some interesting findings. The proportion of commuters from BC that indicated low satisfaction with the transit system’s service in the 2018 calendar year was 65 percent, and the proportion from PEC was 70 percent. These point estimates were based on samples of 5,380 BC commuters and 6,810 PEC commuters, whose system-using commuters number in the millions. (i) In order to compute a valid confidence interval, what values/conditions must be satisfied? (ii) Assume that we may validly construct confidence intervals, and proceed with computing a 95% confidence interval for the difference between the proportions of British Californian commuters and Prince Edward County commuters who report low satisfaction with the public transit system.

Among the answer choices that follow, CHOOSE THE BEST COMBINATIONS OF THE OPTIONS GIVEN IN (i) AND (ii).

a. (i) The product of 5,380 and 65 percent must equal or surpass 10, as must the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized. (ii) (-0.067, -0.033)

b. (i) The product of 6,810 and 70 percent must equal or surpass 10, as must the product of 6,810 and 0.30; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized. (ii) (0.033, 0.067)

c. (i) The product of 5,380 and 65 percent must not surpass 10, true also for the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized. (ii) (-0.066, -0.032)

d. (i) The product of 6,810 and 70 percent must equal or surpass 10, as must the product of 6,810 and 0.30; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and sampling must proceed according to a multi-stage cluster design. (ii) (0.033, 0.067)

e. (i) Each of what we see for (i) in a. and b. is correct. (ii) For (ii), the value given under a. is correct, but the value in b. is incorrect.

Answer 1

(i)

Form given information, $5380\cdot 0.65\cdot 0.35=1223.95$ and $6810\cdot 0.70\cdot 0.30=1430.1$ both are greater than . Samples are independent samples. Since population is large so the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes. Hence, conditions are fulfilled.

(ii)

Here we have m =5380, Ê =0.65 n =6810,2 =0.7 The standard error is: P. (1-ê), P2(1-P2) -0.0086 SE=1 V 17 Level of significance: Critical value: a =0.05 2 =1.96 The requried confidence interval is: (@- ß2)+ SE =(0.65 -0.7)+ (1.96*0.0086) =-0.05 + 0.0169 =(-0.067,-0.033) Excel function for critical value: ENORMSINV(1-(0.05/2)) Hence, the required confidence interval is (-0.067,-0.033).

Correct option is:

a. (i) The product of 5,380 and 65 percent must equal or surpass 10, as must the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized. (ii) (-0.067, -0.033)