Question

A large automobile insurance company selected samples of single and married male policyholders and recorded the number who made an insurance claim over the preceding three-year period.

Single Policyholders	Married Policyholders
n1 = 900	n2 = 400
number making claims = 144	number making claims = 28

(a)

Use α = 0.05. Test to determine whether the claim rates differ between single and married male policyholders.

State the null and alternative hypotheses. (Let p₁ = claim rate for single male policyholders and p₂ = claim rate for married male policy holders.)

H₀: p₁ − p₂ ≥ 0

H_a: p₁ − p₂ < 0

H₀: p₁ − p₂ > 0

H_a: p₁ − p₂ ≤ 0

H₀: p₁ − p₂ ≤ 0

H_a: p₁ − p₂ > 0

H₀: p₁ − p₂ ≠ 0

H_a: p₁ − p₂ = 0

H₀: p₁ − p₂ = 0

H_a: p₁ − p₂ ≠ 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to four decimal places.)

p-value =

State your conclusion.

Reject H₀. We can conclude that there is a difference between claim rates.

Reject H₀. We can not conclude that there is a difference between claim rates.   

Do not reject H₀. We can not conclude that there is a difference between claim rates.

Do not reject H₀. We can conclude that there is a difference between claim rates.

(b)

Provide a 95% confidence interval for the difference between the proportions for the two populations. (Round your answers to four decimal places.)

to

Answer 1

a)

H0: p1 − p2 = 0

Ha: p1 − p2 ≠ 0

p1cap = X1/N1 = 144/900 = 0.16
p1cap = X2/N2 = 28/400 = 0.07
pcap = (X1 + X2)/(N1 + N2) = (144+28)/(900+400) = 0.1323

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.16-0.07)/sqrt(0.1323*(1-0.1323)*(1/900 + 1/400))
z = 4.42

P-value Approach
P-value = 0
As P-value < 0.05, reject the null hypothesis.

Reject H0. We can conclude that there is a difference between claim rates.

b)

Here, , n1 = 900 , n2 = 400
p1cap = 0.16 , p2cap = 0.07

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.16 * (1-0.16)/900 + 0.07*(1-0.07)/400)
SE = 0.0177

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.16 - 0.07 - 1.96*0.0177, 0.16 - 0.07 + 1.96*0.0177)
CI = (0.0553 , 0.1247)