Question

A financial analyst would like to investigate whether the proportion of college graduates who owe more than $50,000 in outstanding student loans is lower in cities than in rural towns. To study this further, the analyst randomly samples 700 college graduates who live in a city and 700 college graduates who live in a rural town. The data from the study are presented in the table below. Let p1 be the proportion of college graduates who live in a city and owe more than $50,000 in outstanding student loans and p2 be the proportion of college graduates who live in a rural town and owe more than $50,000 in outstanding student loans. College graduates who live in a city College graduates who live in a town Number who owe > $50,000 in student loans 155 184 Sample size 700 700 Assume the conditions for the hypothesis test are satisfied. Is there sufficient evidence at α=0.025 to show the proportion of college graduates who owe more than $50,000 in outstanding student loans is lower in cities than in rural towns?

Find p̂ 1,p̂ 2, and p̂ pooled. If needed, round to three decimal places.

Provide your answer below:

proportion 1 =

proportion 2 =

pooled-proportion =

Answer 1

p1cap = X1/N1 = 155/700 = 0.221
p1cap = X2/N2 = 184/700 = 0.263
pooled proportion, pcap = (X1 + X2)/(N1 + N2) = (155+184)/(700+700) = 0.2421

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

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.221-0.263)/sqrt(0.2421*(1-0.2421)*(1/700 + 1/700))
z = -1.83

P-value Approach
P-value = 0.0336
As P-value >= 0.025, fail to reject null hypothesis.