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White remains the most popular car color in the United States, but its popularity appears to be slipping. According to an annual survey by DuPont (Los Angeles Times, February 22,1994 ), white was the color of \(20 \%\) of the vehicles purchased during 1993 , a decline of \(4 \%\) from the previous year. (According to a DuPont spokesperson, white represents "innocence, purity, honesty, and cleanliness.") A random sample of 400 cars purchased during this period in a certain metropolitan area resulted in 100 cars that were white. Does the proportion of all cars purchased in this area that are white appear to differ from the national percentage? Test the relevant hypotheses using \(\alpha=.05\). Does your conclusion change if \(\alpha=.01\) is used?

Step by step solution

01

State the hypotheses

The null hypothesis \(H_0\) states that the proportion of cars that are white does not differ from the national average of \(0.20\), and the alternative hypothesis \(H_1\) states that the proportion of white cars is different from \(0.20\), i.e., \(H_0: p = 0.20\) and \(H_1: p \neq 0.20\).

02

Calculate the Test Statistic

Now we're using the sample data to calculate the test statistic. The test statistic \( z \) for a sample proportion given a null hypothesis can be calculated using the following formula: \( z = (\hat{p} - p_0) / \sqrt{ (p_0 * (1 - p_0)) / n } \). Here, \( \hat{p} = 100/400 = 0.25 \) is the sample proportion, \( p_0 = 0.20 \) is the proportion in the null hypothesis, and \( n = 400 \) is the sample size. Therefore, the test statistic \( z \) equals \( 1.667 \).

03

Find the p-value

The p-value is the probability of observing a sample as extreme as the one we have (or more extreme) given that the null hypothesis is true. Since the hypothesis is two-sided, we should consider both ends of our normal curve. Now we look up the p-value from the z-table, given our z-value of 1.667. The p-value is \( 2*(1 - 0.9525) = 0.095 \).

04

Compare p-value with significance level for \(\alpha = 0.05\)

To draw a conclusion, compare the p-value with the significance level \(\alpha\). If the p-value is less than or equal to \(\alpha\), we reject the null hypothesis. Here, the p-value \((0.095) > \alpha \) (0.05) therefore, we fail to reject the null hypothesis at \(\alpha = 0.05\), which means we do not have sufficient evidence to say the proportion of white cars differs from the national average.

05

Compare p-value with significance level for \(\alpha = 0.01\)

Now compare the p-value with the significance level \(\alpha = 0.01\). Since the p-value is still \((0.095) > \alpha = 0.01\), we fail to reject the null hypothesis at the \(0.01\) significance level, hence we do not have sufficient evidence to conclude the proportion of white cars differs from the national average at a \(0.01\) level of significance.

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