Question

An automobile owner found that 20 years ago, 72% of Americans said that they would prefer to purchase an American automobile. He believes that the number differs from 72% today. He selected a random sample of 43 Americans and found that 38 said that they would prefer an American automobile. Can it be concluded that the percentage today differs from 72%? At α = 0.10, is he correct?

Answer 1

`z=\frac{0.884 -0.72}{\sqrt{(0.72(1-0.72))/(43)}}=2.395`

Now we can find the p value taking incount that we are using a bilateral test

`p_v =2*P(z>2.395)=0.01662`

Since the p value is lower than the significance level of 0.1 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion of Americans that they would prefer an American automobile is significantly different from 0.72 and then the claim is correct

Explanation:

Information given

n=43 represent the random sample taken

X=38 represent the Americans who they would prefer an American automobile

`\hat p=(38)/(43)=0.884` estimated proportion of Americans that they would prefer an American automobile

`p_o=0.72` is the value that we want to check

`\alpha=0.1` represent the significance level

z would represent the statistic

`p_v` represent the p value

Hypothesis to test

We want to verify if the percentage of Americans that they would prefer an American automobile today differs from 72%, then the system of hypothesis are:

Null hypothesis:
`p=0.72`

Alternative hypothesis:
`p \\eq 0.72`

The statistic is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

Replacing the data given we got:

`z=\frac{0.884 -0.72}{\sqrt{(0.72(1-0.72))/(43)}}=2.395`

Now we can find the p value taking incount that we are using a bilateral test

`p_v =2*P(z>2.395)=0.01662`

Since the p value is lower than the significance level of 0.1 we have enough evidence to reject the null hypothesis and we can conclude that the true proportion of Americans that they would prefer an American automobile is significantly different from 0.72 and then the claim is correct