Question

Let n₁=80​, X₁=60​, n₂=80​, and X₂=40.

a. At the 0.01 level of​ significance, is there evidence of a significant difference between the two population​ proportions?

Determine the null and alternative hypotheses.(using "π")

b. Calculate the test​ statistic, Z_STAT, based on the difference p1−p2.

c.Calculate the​ p-value.

d. Determine a conclusion.

______ the null hypothesis. There is ______ evidence to support the claim that there is a significant difference between the two population proportions.

e. Construct a 99​% confidence interval estimate of the difference between the two population proportions.

___≤π₁−π₂≤____

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

b)

p1cap = X1/N1 = 60/80 = 0.75
p2cap = X2/N2 = 40/80 = 0.5
pcap = (X1 + X2)/(N1 + N2) = (60+40)/(80+80) = 0.625

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.75-0.5)/sqrt(0.625*(1-0.625)*(1/80 + 1/80))
z = 3.27

c)

P-value Approach
P-value = 0.0011
As P-value < 0.01, reject the null hypothesis.

d)

Reject the null hypothesis. There is sufficient evidence to support the claim that there is a significant difference between the two population proportions.

e)

Here, , n1 = 80 , n2 = 80
p1cap = 0.75 , p2cap = 0.5

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.75 * (1-0.75)/80 + 0.5*(1-0.5)/80)
SE = 0.074

For 0.99 CI, z-value = 2.58
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.75 - 0.5 - 2.58*0.074, 0.75 - 0.5 + 2.58*0.074)
CI = (0.0591 , 0.4409)

0.0591 ≤π1−π2≤ 0.4409