Question

(a) Two website designs are being compared. 54 students have agreed to be subjects for the study, and they are randomly assigned to visit one or the other of the websites for as long as they like (i.e., half of this number are assigned to visit each of the websites) . For each student the study directors record whether or not the visit lasts for more than a minute. For the first design, 13 students visited for more than a minute; for the second, 6 visited for more than a minute. Find the large-sample 95% confidence interval for the difference in proportions (±0.001). The interval is from to (b) Samples of first-year students and fourth-year students were asked if they were in favor of a new proposed core curriculum. Among the first-year students, 85 said "Yes" and 272 said "No." For the fourth-year students, 115 said "Yes" and 104 said "No.". Find the large-sample 95% confidence interval for the difference in proportions (±0.001). The interval is from to

Answer 1

Answer:

Given that:

a) For the first design, 13 students visited for more than a minute; for the second, 6 visited for more than a minute.

z value at 95% = 1.96

p1 = 13/27 = 0.4815
p2 = 6/27 = 0.2222

CI = ( p1 - p2) +/- z sqrt(p1 (1-p1)/n1 + p2 * (1-p2)/n2)
= ( 0. 4815- 0.2222) +/- 1.96 sqrt(0.4815 (1-0.4815) / 27 + 0.2222 *(1-0.2222)/27)

=0.2593+/-1.96 sqrt (0.4815(0.5185)/27+0.2222(0.7778)/27)

=0.2593+/-1.96 sqrt (0.2497/27+0.1729/27)

=0.2593+/-1.96 sqrt (0.0093+0.0065)

=0.2593+/-1.96 sqrt (0.0158)

=0.2593+/-1.96 (0.1254)

=0.2593+/-0.2456
= (0.0137 , 0.5049)

The 95% CI is (0.0137 , 0.5049)

b)Among the first-year students, 85 said "Yes" and 272 said "No." For the fourth-year students, 115 said "Yes" and 104 said "No.".

z value at 95% = 1.96

p1 = 85/360 =0.2362

p2 = 115/219 = 0.5252

CI = ( p1 - p2) +/- z sqrt(p1 (1-p1)/n1 + p2 * (1-p2)/n2)
= ( 0. 2362 - 0.5252) +/- 1.96 sqrt(0.2362 (1-0.2362) / 360 + 0.5252 *(1-0.5252)/219)

= -0.289 +/- 1.96 sqrt(0.2362(0.7638)/360+0.5252(0.4748)/219)

= -0.289 +/- 1.96 sqrt(0.1805/360+0.2494/219)
= -0.289 +/- 1.96 sqrt(0.0006+0.0012)

= -0.289 +/- 1.96 sqrt(0.0018)

= -0.289 +/- 1.96 (0.0417)

=-0.289 +/-0.08174

= (-0.3708, -0.2073)

The 95% CI is (-0.3708 , -0.2073)