Question

proponents of new milk dairy on the south

Proponents of a new milk dairy on the south side of town claim at least half of the people who live in that area are in favor of the dairy. A group of residents in the area that formed to fight the dairy project claim the estimate is unrealistic and vastly overestimates the proportion of residents in favor of the dairy. The city government then hired an independent firm to conduct a survey. The survey found 277 of 642 residents randomly contacted in the area were in favor of the dairy. (a) What assumptions are necessary and how are they satisfied in order to complete the desired test?

Answer 1

Part a

The assumption required to this test is given as the sample should be come from normally distributed population and sample size should be enough. For this survey the sample size is given as 642 which is adequate for using the test for this claim.

Part b

Here, we have to use z test for population proportion.

H₀: p ≥ 0.5 versus H_a: p < 0.5

This is a lower tailed test.

We assume α = 0.05

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 277

n = sample size = 642

p̂ = x/n = 277/642 = 0.431464174

p = 0.5

q = 1 - p = 0.5

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.431464174 – 0.5)/sqrt(0.5*0.5/642)

Z = -3.4731

P-value = 0.0003

(by using z-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is insufficient evidence to conclude that at least half of the people who live in that area are in favour of the dairy.