Answer to Question #151897 in Statistics and Probability for Lima

Question

A simple random sample of size n drivers were asked if they drive a car manufactured in a certain country. Of the drivers​ surveyed, responded that they did. Determine if more than half of all drivers drive a car made in this country at the level of significance. Complete parts ​(a) through ​(c).
​(a) Determine the null and alternative hypotheses.
​:
p
equals

​:
p
greater than
​(b) State the conclusion for the test.
Choose the correct answer below.
A.
because the​ P-value is the level of significance.
B.
because the​ P-value is the level of significance.
C.
because the​ P-value is the level of significance.
D.
because the​ P-value is the level of significance.
​(c) State the conclusion in context of the problem.

Answer 1

a) The following null and alternative hypotheses need to be tested:

$$H_0: p\leq0.5$$

$$H_1: p>0.5$$

b) The z-statistic is computed as follows:

$$z=\dfrac{\bar{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$$

Let $n=200, x=115$

$$\bar{p}=\dfrac{x}{n}=\dfrac{115}{200}=0.575$$

Then

$$z=\dfrac{\bar{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$$

$$z=\dfrac{0.575-0.5}{\sqrt{\dfrac{0.5(1-0.5)}{200}}}\approx2.1213$$

Using the P-value approach: The p-value is $p=P(Z>2.1213)=0.016948,$ and since $p=0.016948<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

c) There is enough evidence to claim that the population proportion $p$ is greater than $p_0=0.5,$ at the $\alpha=0.05$  significance level.