Stats module 6 homework question

Question

A poll reported that 31% of 130 Canadians between the ages of 25 and 29 had started saving money for retirement. Of the 140 Canadians surveyed between the ages of 21 and 24, 25% had started saving for retirement. Carry out an appropriate hypothesis test and see whether there is any difference between the proportions of Canadians between the ages of 25 and 29 and the ages of 21 and 24 who had started saving for retirement.

(a) What are the null and alternative hypotheses?

H₀: pμp1 - p2μ1 - μ2μd <≠>= 0 or 0.06

H_A: pμp1 - p2μ1 - μ2μd =≤≥<≠> 0 or 0.06

(b) What is the test statistic? (Round your answer to 2 decimal places, if needed.)

(c) Using the statistical table, what is the p-value? (Round your answer to 4 decimal places, if needed.)

(d) Based on the p-value, what can you conclude?

 There is about a 27.13% chance that the two proportions are equal.

 If there is no difference in the proportions, there is about a 13.56% chance of seeing the observed difference or larger (as extreme or more extreme) by natural sampling variation.

 There is about a 27.13% chance that the two proportions are unequal.

 If there is no difference in the proportions, there is about a 13.56% chance of seeing the exact observed difference by natural sampling variation.

 If there is no difference in the proportions, there is about a 27.13% chance of seeing the observed difference or larger (as extreme or more extreme) by natural sampling variation.

Answer 1

I can give you the python code to address this question:

from scipy.stats import norm

# Defining given values

n1, n2 = 130, 140 # Sample sizes

p1_hat, p2_hat = 0.31, 0.25 # Sample proportions

# Calculate the pooled proportion

p_hat_pooled = ((p1_hat * n1) + (p2_hat * n2)) / (n1 + n2)

# Standard error for the difference in proportions

SE = (p_hat_pooled * (1 - p_hat_pooled) * (1 / n1 + 1 / n2)) ** 0.5

# Test statistic (z)

z_stat = (p1_hat - p2_hat) / SE

p_hat_pooled, SE, z_stat

# Calculate the p-value for a two-tailed test

p_value = 2 * (1 - norm.cdf(abs(z_stat)))

p_value