Question

9. A company wants to know if the proportion of consumers in Houston that recognizes its brand name is higher than the proportion of consumers in Austin that recognizes its brand name. In a random sample of consumers in Houston, 406 out of 567 consumers recognized the company’s brand name. In a random sample of consumers in Austin, 423 out of 644 consumers recognized the company’s brand name. Make Houston group 1 and make Austin group 2.

(a) What is the null hypothesis and what is the alternative hypothesis?

(b) What is the sample proportion for group 1? (round to 5 digits after the decimal place)

(c) What is the sample proportion for group 2? (round to 5 digits after the decimal place)

(d) What is the pooled estimator for p? (round to 5 digits after the decimal place)

(e) What is the standard error for the difference in the sample proportions? (Use ˜σp1−p2 and round to 5 digits after the decimal place.)

(f) What is the value of the test statistic? (Round to 2 digits after the decimal place.)

(g) What is the p-value of the test? (Round to 3 digits after the decimal place.)

(h) Do we reject or not reject the null hypothesis at the .05 level of significance? Reject Not reject

(i) Can we interpret the difference in the population proportions as a causal effect? Yes, it has a causal interpretation. No, it does not have a causal interpretation.

Answer 1

a)

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

b)

p1cap = X1/N1 = 406/567 = 0.71605

c)

p1cap = X2/N2 = 423/644 = 0.65683

d)

pcap = (X1 + X2)/(N1 + N2) = (406+423)/(567+644) = 0.68456

e)
Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.71605 * (1-0.71605)/567 + 0.65683*(1-0.65683)/644)
SE = 0.02662

f)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.71605-0.65683)/sqrt(0.68456*(1-0.68456)*(1/567 + 1/644))
z = 2.21

g)

P-value Approach
P-value = 0.014

h)

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

i)
Yes, it has a causal interpretation.