Question

In a survey of 1,005 adult Americans, \(46 \%\) indicated that they were somewhat interested or very interested in having Web access in their cars (USA Today, May 1,2009 ). 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 \(0.50 .\) 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 representative ofp adult Americans.

Short answer

Whether the manager's claim is correct will depend on the test statistic and the P-value. If the calculated P-value is less than 0.05, then the manager's claim would be incorrect and it can be statistically concluded that less than 50% of adult Americans want car web access.

Step-by-step solution

Set Up the Hypothesis

To test the manager's claim statistically, the first step is to set up null and alternative hypotheses. The null hypothesis (H0) would be that the population proportion (P) is equal to or greater than 0.50. The alternative hypothesis (H1) would be that P is less than 0.50.

Calculate the Test Statistic

Once the hypotheses are established, the next step is to perform a one-sample Z-test for proportions. Given that we have a large sample size (n > 1000), it is valid to apply the Central Limit Theorem and use the z-distribution. The formula for the test statistic (Z) is \(Z = (p̂ - P) / \sqrt{(P(1 - P) / n)}\) where p̂ is the sample proportion (0.46), and n is the sample size (1005).

Find the P-Value

The third step is to find the P-value associated with the observed test statistic. The P-value is the probability that a z-score is at least as extreme as the one calculated from the sample data (in the direction of the alternative hypothesis). If the P-value is small (typically ≤ 0.05), reject the null hypothesis in favor of the alternative hypothesis.

Make a Decision

Based on the P-value obtained in the previous step, make a decision about the null hypothesis. If the P-value is small, reject the null hypothesis and accept that the true population proportion is less than 0.50. If the P-value is large, fail to reject the null hypothesis and the manager's claim stands.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial in hypothesis testing. The null hypothesis (\(H_0\)) represents the status quo or a statement of no effect. For example, it might assert that the population proportion is a specified value. In the exercise about web access interest in cars, the null hypothesis posits that the population proportion (\(P\text{\textsubscript{population}}\text{\textsubscript{proportion}}\)) who want web access in their cars is at least 50%.

In contrast, the alternative hypothesis (\(H_1\text{\textsubscript{or}} H_A\text{\textsubscript{alternative}} \text{\textsubscript{hypothesis}}\)) presents the researcher's belief which is contrary to the null hypothesis. This hypothesis is what you aim to support with evidence from the data. In our case, it suggests that the actual interest is lower than 50%. The formulation of these two hypotheses is a critical first step in the hypothesis testing process and provides the framework for using statistical methods to make inferences about a population.

One-Sample Z-Test

The one-sample Z-test is a statistical method used to determine whether the sample evidence is strong enough to reject the null hypothesis for a single population proportion. This test is appropriate when the sample size is large and the sample can be assumed to come from a normally distributed population, a condition satisfied by the Central Limit Theorem when sample sizes are typically over 30.

In the educational exercise presented, we have a significant sample size (n=1005), which validates the use of the Z-test. The test statistic is calculated using the formula \[\begin{equation}Z = (\hat{p} - P) / \sqrt{(P(1 - P) / n)}\end{equation}\]where \(\hat{p}\) is the sample proportion and \(P\) is the hypothesized population proportion. The calculated Z-value tells us how many standard deviations our sample proportion is from the hypothesized population proportion. The next step would be to compare this value to a standard normal distribution to determine the P-value.

P-Value

The P-value is a pivotal concept in hypothesis testing, representing the probability that the observed data, or more extreme, could occur if the null hypothesis is true. It's a measure of the strength of the evidence against the null hypothesis.

If the P-value is low (commonly below 0.05), it indicates that such an extreme observed result would be very unlikely if the null hypothesis were true. This outcome leads statisticians to reject the null hypothesis in favor of the alternative hypothesis. Conversely, a large P-value suggests that the observed result is not particularly unusual under the null hypothesis. In the context of our web access interest case, a small P-value would imply that the true population proportion of interest is likely less than 50%, challenging the marketing manager's claim.

Population Proportion

The population proportion is a key parameter in statistics that represents the fraction of the population of interest who exhibit a particular feature – it's denoted as \(P\) in our formulas. For instance, it might denote the percentage of people who want web access in their cars.

The sample proportion (\(\hat{p}\)), observed from the survey or experiment, serves as an estimate of this unknown population proportion. The accuracy of this estimate depends on the sample size and the variability in the population.

If the sample size is large and randomly selected, it's more likely that the sample proportion will be close to the true population proportion. As observed in the given example, where 46% of a large sample of 1005 adult Americans showed interest in web access for cars, we infer about the overall population's preference based on this sample. To decide whether the sample proportion significantly differs from the claimed population proportion, we use hypothesis testing methods such as the one-sample Z-test described previously.