Question

Which preconditions have to be fulfilled that I can apply a z-Test here? Is it 0.47*125≥5 and (1-0.47)*125≥5 OR 0.44*125≥5 and (1-0.44)*125≥5 OR BOTH?

Environmental health indicators include air quality, water quality, and food quality. Suppose

that 25 years ago, 47% of food samples contained pesticide residues (i.e. take this as a

population value). In a recent study, 44 of 125 food samples contained pesticide residues.

State the hypotheses that can be used to test whether the population proportion has

declined. Use a = 0.01. What is your conclusion? … I am only interested in preconditions TIA

Answer 1

Ans:

$H_0:p=0.47$

$H_a:p<0.47$

Hypothised population proportion=p₀=047

n=125

Pre-conditions for z-test:

np₀>=5 and n(1-p₀)>=5

125*0.47>=5 and 125*(1-0.47)>=5

sample proportion=44/125=0.352

Test statistic:

z=(0.352-0.47)/sqrt(0.47*(1-0.47)/125)

z=-2.64

p-value=P(z<-2.64)=0.0041

As,p-value<0.01,we reject the null hypothesis.

There is sufficient evidence to conclude that the population proportion has declined.