Answer to Question #238244 in Statistics and Probability for suvi

Question

A farming cooperative in the KwaMashu buys wheat seeds for its farmer members from seed

merchants. A particular seed merchant claims that their wheat seeds have at least an 80%

germination rate. Before the farming cooperative will buy from this seed merchant, they want to

verify this claim. A random sample of 320 wheat seeds supplied by this seed merchant was tested,

and it was found that only 230 seeds germinated. Is there sufficient statistical evidence at the 3%

significance level to justify the purchase of wheat seeds from this seed merchant? Use the p-value

approach to conduct a hypothesis test for a single proportion, and report the findings to the

KwaMashu farming cooperative. (10)

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p\geq0.8$

$H_1:p<0.8$

This corresponds to a left-tailed test, for which a z-test for one population proportion will be used.

Based on the information provided, the significance level is $\alpha=0.03,$ and the critical value for a left-tailed test is $z_c=-1.8808.$

The rejection region for this left-tailed test is $R=\{z:z<-1.8808\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{\dfrac{230}{320}-0.8}{\sqrt{\dfrac{0.8(1-0.8)}{320}}}=-3.6336$$

Using the P-value approach: The p-value is $p=P(Z<-36336)\approx0.000140,$ and since $p=0.000140<0.03=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population proportion $p$  is less than 0.8, at the $\alpha=0.03$ significance level.