91Ó°ÊÓ

Refer to the following setting. The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?" Here are her data: $$ \begin{array}{lcccc} \hline \text { Type of Food: } & \text { Asian } & \text { Mexican } & \text { Pizza } & \text { Hamburgers } \\ \text { Count: } & 18 & 22 & 39 & 21 \\ \hline \end{array} $$ The \(P\) -value for a chi-square test for goodness of fit is \(0.0129 .\) Which of the following is the most appropriate conclusion? (a) Because 0.0129 is less than \(\alpha=0.05\), reject \(H_{0}\). There is convincing evidence that the food choices are equally popular. (b) Because 0.0129 is less than \(\alpha=0.05\), reject \(H_{0}\). There is not convincing evidence that the food choices are equally popular. (c) Because 0.0129 is less than \(\alpha=0.05\), reject \(H_{0}\). There is convincing evidence that the food choices are not equally popular. (d) Because 0.0129 is less than \(\alpha=0.05,\) fail to reject \(H_{0}\). There is not convincing evidence that the food choices are equally popular. (e) Because 0.0129 is less than \(\alpha=0.05\), fail to reject \(H_{0}\). There is convincing evidence that the food choices are equally popular.

Step by step solution

01

Understand the Hypotheses

Before performing a chi-square test for goodness of fit, identify the null and alternative hypotheses. \( H_0 \) (the null hypothesis) assumes that all food type preferences are equally popular among students. \( H_a \) (the alternative hypothesis) suggests that the preferences are not equally popular.

02

Identify the Significance Level

The problem specifies a significance level \( \alpha = 0.05 \). This value will be used to determine whether or not to reject the null hypothesis. If the \( p \)-value is less than \( \alpha \), \( H_0 \) is rejected.

03

Analyze the P-value

The \( p \)-value given is 0.0129. Compare this \( p \)-value against the significance level \( \alpha = 0.05 \). Since 0.0129 is less than 0.05, the null hypothesis \( H_0 \) can be rejected.

04

Determine the Conclusion

Rejecting \( H_0 \) implies there is convincing evidence that the food choices are not equally popular. Therefore, option (c) is the correct conclusion: "Because 0.0129 is less than \( \alpha=0.05 \), reject \( H_0 \). There is convincing evidence that the food choices are not equally popular."

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In the realm of statistics, the null hypothesis, often denoted as \( H_0 \), represents a default assumption that there is no effect or no difference present in your data. In the context of the chi-square test for goodness of fit, it implies that all categories—in this case, food preferences like Asian, Mexican, pizza, and hamburgers—are equally popular. The null hypothesis is like saying, "I expect nothing special is going on."

The importance of stating a null hypothesis lies in providing a starting framework for statistical testing. By assuming no difference or effect, we have a baseline to test against. When the evidence, as expressed through data, suggests otherwise, the null hypothesis can be rejected. Rejection of \( H_0 \) indicates that the observed data likely did not occur by random chance alone and that other factors could influence the results.

P-Value

The concept of a p-value is a fundamental idea in hypothesis testing. A p-value quantifies the probability of obtaining a test result that is at least as extreme as the one observed, under the assumption that the null hypothesis is true. It helps in determining how well the sample data supports a hypothesis.

In this exercise, the p-value is given as 0.0129. This indicates the likelihood of observing the given distribution of food preferences if, in reality, students had no strong preference for any specific type of food (the null hypothesis is true).

• If the p-value is very low, it suggests that the sample data significantly deviates from what was expected under \( H_0 \).
• A common threshold for "very low" is 0.05, meaning there is less than a 5% probability that the observed results occurred under \( H_0 \).

Here, since 0.0129 is less than 0.05, it supports rejecting the null hypothesis, suggesting that food preferences are not equally distributed among students.

Significance Level

The significance level, denoted by \( \alpha \), acts as a threshold for decision-making in hypothesis testing. It signifies the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. Choosing \( \alpha \) is a balance; a lower \( \alpha \) means stricter testing and fewer false positives, but also reduces sensitivity to true differences.

Typically, a common choice for the significance level is 0.05, meaning there is a 5% chance of mistakenly rejecting the null hypothesis when it is true. In this test, the significance level is set at \( \alpha = 0.05 \).

• A p-value less than \( \alpha \) leads to rejecting \( H_0 \).
• A p-value greater than \( \alpha \) implies insufficient evidence to reject \( H_0 \).

By setting an acceptable risk level, the significance level ensures that decisions based on statistical tests are reliable.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_a \), proposes that there is an effect or a difference present in the data. In the case of the chi-square test for goodness of fit, \( H_a \) suggests that at least one category has a preference different from others. It acts as the counterpart of the null hypothesis.

While the null hypothesis assumes no change or effect, the alternative hypothesis is what researchers typically hope to support. To accept \( H_a \), the data must provide significant evidence against \( H_0 \).

• If the p-value is lower than the significance level, it indicates stronger evidence for the alternative hypothesis.
• The decision to reject \( H_0 \) and support \( H_a \) indicates that the results are not due to random chance.

In this exercise, rejecting \( H_0 \) concluded that students' food preferences are not equally popular, affirming the alternative hypothesis.