Question

In a survey of 1005 adult Americans, \(46 \%\) indicated that they were somewhat interested or very interested in having web access in their cars (USA Today, May I. 2009 ). Suppose that the marketing manager of a car manufacturer claims that the \(46 \%\) is based only on a sample and that \(46 \%\) is close to half, so there is no reason to believe that the proportion of all adult Americans who want car web access is less than \(.50 .\) Is the marketing manager correct in his claim? Provide statistical evidence to support your answer. For purposes of this exercise, assume that the sample can be considered as representative of adult Americans.

Short answer

The final answer to this problem will be based on the calculated p-value. Since the calculation of this p-value requires access to statistical tables or software, it cannot be precisely determined within this context. However, if the p-value is less than 0.05, the marketing manager's claim will be considered incorrect. If the p-value is greater than 0.05, there is insufficient evidence to refute the marketing manager's claim.

Step-by-step solution

Define Hypotheses

The first step in testing the marketing manager's claim is to define the null and alternative hypotheses. In this case, the null hypothesis (H0) would be that the population proportion is equal to 0.50, i.e., H0: p= 0.50. The alternative hypothesis (H1) would be that the population proportion is less than 0.50, i.e., H1: p < 0.50.

Test Statistic

The test statistic can be calculated using the formula: Z = (p̂ - p0) / sqrt((p0(1 - p0))/n), where p̂ is the sample proportion, p0 is the population proportion given by H0, and n is the sample size. Substituting the given values, we get Z = (0.46-0.50) / sqrt((0.50*(1 - 0.50))/1005).

Determine the p-value

The p-value associated with a given Z value reports the likelihood of finding a statistic as extreme as, or more extreme than, the observed statistic under the null hypothesis. It can be found using Z-tables or software. If the p-value is less than the significance level (usually chosen as 0.05), we reject the null hypothesis.

Conclusion

If the calculated p-value is less than the chosen significance level, we reject the null hypothesis and assert that the marketing manager's claim is not correct. Alternatively, if the p-value is greater than the significance level, we fail to reject the null hypothesis, suggesting that the marketing manager's claim could be correct.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that implies no effect or no difference in the population. It is essentially a baseline against which we measure our results. For our exercise, the null hypothesis (denoted as \(H_0\)) suggests that the true population proportion of adult Americans who want car web access is \(0.50\).
This represents the claim that the proportion is similar enough to half, showing no significant difference from the proverbial midpoint.
In mathematical terms:

• \(H_0: p = 0.50\)

Our goal is to test this claim against the evidence gathered from the sample.

Alternative Hypothesis

The alternative hypothesis provides the opposite stance to the null hypothesis. It proposes that there is a significant effect or difference. When conducting a hypothesis test, we wish to see if the data provides enough evidence to support this alternative perspective.
In our given example, the alternative hypothesis (denoted as \(H_1\) or \(H_a\)) suggests that the true population proportion is less than \(0.50\), contradicting the marketing manager's statement.
Mathematically, this is expressed as:

• \(H_1: p < 0.50\)

This suggests that less than half of adult Americans want web access in their cars.

P-value

The p-value plays a crucial role in hypothesis testing, as it helps quantify the evidence against the null hypothesis. It provides the probability of observing the test results under the assumption that the null hypothesis is true.
A smaller p-value indicates stronger evidence in favor of the alternative hypothesis. In decision-making, we compare the p-value to a chosen significance level (usually \(0.05\)):

• If \(\text{p-value} < 0.05\), we reject the null hypothesis.
• If \(\text{p-value} \geq 0.05\), we fail to reject the null hypothesis.

In this exercise, the p-value will help us determine if the difference between the surveyed and claimed proportions is statistically significant.

Test Statistic

The test statistic is a standard value we calculate from our sample data, which follows a certain distribution under the null hypothesis. For proportion tests, like in this exercise, we often use the Z-statistic, which tells us how far our sample proportion is from the assumed population proportion under the null hypothesis.
The formula for the Z-statistic in testing proportions is:

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

where:

• \(\hat{p}\) is the sample proportion (\(0.46\) in this case).
• \(p_0\) is the population proportion under the null hypothesis (\(0.50\)).
• \(n\) is the sample size (1005).

Computing this will give us our test statistic, which we use to find our p-value.

Population Proportion

Population proportion is a parameter that indicates the fraction of the population that shares a certain characteristic. In hypothesis testing, we often compare the observed sample proportion (\(\hat{p}\)) with an expected population proportion (\(p\)) based on past claims or analysis.
In our example, the survey suggests that \(46\%\) of adult Americans are interested in car web access, while the hypothesized population proportion (from the null hypothesis) is \(50\%\).
Understanding and comparing these proportions through hypothesis tests helps in verifying claims about the population and provides insights into whether the observed sample accurately reflects the broader population trend.