Questions: A publisher reports that 44% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually above the reported percentage. A random sample of 340 found that 50% of the readers owned a particular make of car. Is there sufficient evidence at the 0.05 level to support the executive's claim? State the conclusion of the hypothesis test.

Solution Steps

We want to test the claim that the percentage of readers who own a particular make of car is greater than the reported percentage of \(44\%\).

• Null Hypothesis (\(H_0\)): \(p \leq 0.44\)
• Alternative Hypothesis (\(H_a\)): \(p > 0.44\)

The test statistic for the sample proportion is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

• \(\hat{p} = 0.50\) (sample proportion)
• \(p_0 = 0.44\) (hypothesized population proportion)
• \(n = 340\) (sample size)

We find:

\[ Z = \frac{0.50 - 0.44}{\sqrt{\frac{0.44(1 - 0.44)}{340}}} = 2.2288 \]

The P-value associated with the test statistic \(Z = 2.2288\) is calculated to be:

\[ \text{P-value} = 0.0129 \]

For a significance level of \(\alpha = 0.05\) in a one-tailed test, the critical value is:

\[ Z_{critical} = 1.6449 \]

We compare the P-value to the significance level:

• Since \(0.0129 < 0.05\), we reject the null hypothesis.

Final Answer