In this exercise we examine the effect of the sample size on the significance te

Question

In this exercise we examine the effect of the sample size on the significance test for comparing two proportions. In each case suppose that p1 = 0.65 and p2 = 0.45, and take n to be the common value of n1 and n2. Use the z statistic to test H0: p1 = p2 versus the alternative Ha: p1 p2. Compute the statistic and the associated P-value for the following values of n: 40, 50, 60, 80, 380, 480, and 980. Summarize the results in a table. (Test the difference p1 p2. Round your values for z to two decimal places and round your P-values to four decimal places.)

n z P-value

40

50

60

80

380

480

980

Question 2

We have studied the effect of the sample size on the margin of error of the confidence interval for a single proportion. In this exercise we perform some calculations to observe this effect for the two-sample problem. Suppose that p1 = 0.7 and p2 = 0.5, and n represents the common value of n1 and n2. Compute the 95% margins of error for the difference between the two proportions for n = 50, 60, 70, 90, 390, 490, and 990. Present the results in a table. (Give the large-sample margins of error. Round your answers to three decimal places.)

Explanation / Answer

Q1.

question1 = function(p1, n1, p2, n2)
{
table = matrix(c(p1*n1, p2*n2, (1-p1)*n1, (1-p2)*n2), nrow = 2, ncol=2)
r = prop.test(x = table, conf.level = 0.95, correct = FALSE)
z = sqrt(r$statistic)
p = r$p.value
answer1 = unname(c(z, p))
print(answer1)
}

> question1(p1 = 0.65, p2 = 0.45, n1 = 40, n2 = 40)
[1] 1.7978663 0.0721982
> question1(p1 = 0.65, p2 = 0.45, n1 = 50, n2 = 50)
[1] 2.01007563 0.04442318
> question1(p1 = 0.65, p2 = 0.45, n1 = 60, n2 = 60)
[1] 2.20192753 0.02767043
> question1(p1 = 0.65, p2 = 0.45, n1 = 80, n2 = 80)
[1] 2.54256690 0.01100415
> question1(p1 = 0.65, p2 = 0.45, n1 = 380, n2 = 380)
[1] 5.541396 3.000696e-08
> question1(p1 = 0.65, p2 = 0.45, n1 = 480, n2 = 480)
[1] 6.227992 4.724525e-10
> question1(p1 = 0.65, p2 = 0.45, n1 = 980, n2 = 980)
[1] 8.898984 5.636013e-19

Q2.

question2 = function(p1, n1, p2, n2)
{
table = matrix(c(p1*n1, p2*n2, (1-p1)*n1, (1-p2)*n2), nrow = 2, ncol=2)
r = prop.test(x = table, conf.level = 0.95, correct = FALSE)
answer2 = (r$conf.int[2] - r$conf.int[1])/2
print(answer2)
}

> question2(p1 = 0.7, p2 = 0.5, n1 = 50, n2 = 50)
[1] 0.1879931
> question2(p1 = 0.7, p2 = 0.5, n1 = 60, n2 = 60)
[1] 0.1716135
> question2(p1 = 0.7, p2 = 0.5, n1 = 70, n2 = 70)
[1] 0.1588832
> question2(p1 = 0.7, p2 = 0.5, n1 = 90, n2 = 90)
[1] 0.1401218
> question2(p1 = 0.7, p2 = 0.5, n1 = 390, n2 = 390)
[1] 0.06731234
> question2(p1 = 0.7, p2 = 0.5, n1 = 490, n2 = 490)
[1] 0.06005221
> question2(p1 = 0.7, p2 = 0.5, n1 = 990, n2 = 990)
[1] 0.04224832

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