Problem 48 - Basic Statistics for Business and Economics with Student

Question

According to a study by the American Pet Food Dealers Association, 63 percent of U.S. households own pets. A report is being prepared for an editorial in the San Francisco Chronicle. As a part of the editorial a random sample of 300 households showed 210 own pets. Does this data disagree with the Pet Food Dealers Association data? Use a .05 level of significance.

Step-by-Step Solution

Answer

Yes, the data disagrees; the sample proportion of 70% significantly differs from the 63% reported.

Step 1: Define the Hypotheses

First, we need to define the null hypothesis and the alternative hypothesis. The null hypothesis (\(H_0\)) will assert that the proportion of households owning pets according to the sample is equal to the proportion reported by the association, which is 63% or \(p_0 = 0.63\). The alternative hypothesis (\(H_a\)) will assert that these proportions are not equal, \(p eq p_0\).

Step 2: Calculate the Test Statistic

The test statistic for proportions is computed using the formula \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion, and \(n\) is the sample size. In this situation, \(\hat{p} = \frac{210}{300} = 0.70\), \(p_0 = 0.63\), and \(n = 300\). Substitute these values to find \(z\).

Step 3: Determine the Critical Value and Compare

For a significance level of 0.05 in a two-tailed test, the critical z-values are \(\pm1.96\). Calculate the test statistic. If it falls outside of -1.96 to 1.96, we will reject the null hypothesis.

Step 4: Conclusion

Calculate the test statistic: \(z = \frac{0.70 - 0.63}{\sqrt{\frac{0.63 \times 0.37}{300}}}\). This gives \(z \approx 2.61\). Since 2.61 is greater than 1.96, we reject the null hypothesis at the 0.05 level of significance. This suggests that the sample data significantly disagrees with the proportion of U.S. households owning pets as given by the American Pet Food Dealers Association.

Key Concepts

Test Statistic

The test statistic is a pivotal element in hypothesis testing. It is used to determine whether to reject the null hypothesis, thus guiding our decision-making process. In this scenario, we are dealing with proportions, specifically, examining the proportion of households owning pets. When calculating the test statistic for proportions, we use the following formula:

• \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)

Here,

• \(\hat{p}\) is the sample proportion.
• \(p_0\) is the population proportion as proposed by the null hypothesis.
• \(n\) is the sample size.

In this exercise, the sample proportion (\(\hat{p}\)) is 0.70, obtained by dividing the number of households that own pets by the total sampled households (210/300). The assumed population proportion (\(p_0\)) is 0.63. By substituting these values into the formula, we compute the test statistic to be approximately 2.61. This statistic will later be used to assess the validity of the null hypothesis based on its comparison to critical values.

Significance Level

The significance level, commonly denoted as \(\alpha\), plays a crucial role in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. Essentially, it sets the threshold for what is considered statistically significant. A typical significance level in many studies is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none. This level is a balance between being too lenient and too stringent.
In our example, a 0.05 significance level is used to evaluate the test. For a two-tailed test, this means we look for critical values at both ends of the distribution. Typically, for \(\alpha = 0.05\), the critical z-values are \(\pm1.96\). If the calculated test statistic falls beyond these, it indicates a significant difference, prompting the rejection of the null hypothesis. Thus, the significance level serves as our standard for making decisions based on the calculated test statistic, ensuring our findings are not mere flukes of chance.

Null Hypothesis

The null hypothesis (\(H_0\)) is a fundamental component of hypothesis testing, representing the default assumption that there is no effect or difference. It provides a statement that can be tested statistically, acting as the "status quo" that we challenge. In our example, the null hypothesis states that the proportion of pet-owning households in our sample equals the population proportion reported by the American Pet Food Dealers Association, specifically, 63%. This can be expressed as:

• \(H_0: p = 0.63\)

The purpose of testing the null hypothesis is to evaluate whether any observed difference (such as the sample proportion being 0.70 instead of 0.63) is significant enough to suggest that \(H_0\)should be rejected. By determining the test statistic and comparing it to critical values, we can decide to reject or fail to reject the null hypothesis based on statistical evidence. In this instance, because the test statistic leads us to reject \(H_0\), it suggests that the sample data significantly differs from the population proportion, indicating potential discrepancies with the established statistic from the association.