Chapter 20, Problem 10E

To determine

To correct the mistakes you find in a student’s attempt to test an appropriate hypothesis.

Explanation of Solution

It is given in the question that the newsletter reported that $90\%$ of adults drink milk. A regional farmers’ organization planning a new marketing campaign across its multicounty area polls a random sample of $750$ adults living here. Thus, the student conducted an appropriate hypothesis for this context. So, the mistakes made by the student is explained as,

Firstly, we are looking at true proportion, not sample for the hypotheses. Alternative hypothesis should be two-sided since the question asks "not accurate". So, the hypothesis will be:

  $\begin{array}{l}{H}_0:p=0.9\\ H_a:p\ne 0.9\end{array}$

Secondly, the check of the normalcy of the sample, will be as:

  $\begin{array}{l}np=0.9(750)>10\\ n(1-p)=0.10(750)>10\end{array}$

Thirdly, the true proportion is used to find standard deviation, not sample proportion, which is as:

  $\begin{array}{l}SD=\sqrt{\frac{0.9(0.10)}{750}}\\ =0.011\end{array}$

And, z -score is calculated by subtracting the true proportion from the sample proportion and dividing by standard deviation, which is:

  $\begin{array}{l}z=\frac{0.876-0.9}{0.011}\\ =-2.18\end{array}$

So, which gives,

  $P(z<-2.18)=0.029$

Which implies, since P-value is less than $0.05$ , we reject the null hypothesis. And we reject that the proportion of adults who drink milk is $90\%$ .