Question

A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have DVD players.
If you were to conduct a test to determine whether there is evidence that the proportion is different from 0.25,

a.) State the null hypothesis

b.) State the alternative hypothesis

c.) What critical values should be used?

d.) What is the test statistic?

e.) What is the p-value?

f.) Interpret th emeaning of the p-value

g.) Should the null hypothesis be rejected?

Answer 1

Answer with explanation:

Let p be the population proportion .

Then, $p=\dfrac{5000}{20000}=0.25$

According to the given information, we have

$a)\ \text{Null hypothesis }H_0: p=0.25\\\\b)\ \text{Alternative hypothesis} H_a: p\neq0.25$, since alternative hypothesis is two-tailed , so the hypothesis test is a two-tail test.

Since sample size is large (n> 30), we use z-test.

Let us consider 95% confidence i.e.$\alpha=0.05$.

c) Critical value = $z_{\alpha/2}=\pm1.96$

Sample proportion : $\hat{p}=\dfrac{96}{300}=0.32$

d) Test statistic : $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

i.e. $z=\dfrac{0.32-0.25}{\sqrt{\dfrac{0.25(0.75)}{300}}}=2.8$

e) P-value (for two-tail) : $2(Pz>2.8)=0.0051103$

f) P-value is the probability value that we have falsely rejected the null hypothesis.

g) Since observed z-value (2.8) does not lie in critical interval (-1.96, 1.96), it means the null hypothesis is rejected.