Question

An article states that at least 1.8% of all soil on farms is contaminated. A researcher wishes to challenge this claim and gain evidence that the true figure is lower than this. Therefore he obtains data from 297 probes, of which five detected the soil as contaminated.

a) State the null and alternative hypotheses.

b) Show the areas of rejection and non-rejection in an appropriate diagram.

c) Test the null hypothesis to the 95% confidence level and state your conclusions. Show all your working.

d) If your conclusion were to turn out to be incorrect, state the type of error made and why.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

a)

Null hypothesis: P = 0.018
Alternative hypothesis: P < 0.018

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

b)

Rejection region is z < - 1.645

S.D = 0.0077146
z = (p - P) / S.D

z = - 0.151

where P is the hypothesised value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

c) Interpret results. Since the z-value (- 0.151) does not lies in the rejection region, hence we have to accept the null hypothesis.

d) From the above test we have sufficient evidence in the favor of the claim that at least 1.8% of all soil on farms is contaminated.