Question

The USDA is concerned that most farmers are older Americans. In 2000, it was reported that 25% of all US farmers were over the age of 65. In 2020, a random sample of 350 US farmers was taken and it was found that 96 were over the age of 65. The USDA claims that the proportion of all farmers who are over the age of 65 is now greater than 25%. Does this provide evidence to support the USDA’s claim at the 5% significance level? Run a hypothesis test. Be sure state Ho and Ha, the test statistic and p-value, whether you reject Ho or not and your conclusion in terms of the claim.

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.25
Alternative Hypothesis, Ha: p > 0.25

Rejection Region
This is right tailed test, for α = 0.05
Critical value of z is 1.64.
Hence reject H0 if z > 1.64

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.2743 - 0.25)/sqrt(0.25*(1-0.25)/350)
z = 1.05

P-value Approach
P-value = 0.1469
As P-value >= 0.05, fail to reject null hypothesis.

there is not sufficient evidence to conclude that the proportion is greater than 25%