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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: $$ \begin{array}{lcccc} \hline \text { Type of Food: } & \text { Asian } & \text { Mexican } & \text { Pizza } & \text { Hamburgers } \\ \text { Count: } & 18 & 22 & 39 & 21 \\ \hline \end{array} $$ (a) \(\frac{(18-25)^{2}}{25}+\frac{(22-25)^{2}}{25}+\frac{(39-25)^{2}}{25}+\frac{(21-25)^{2}}{25}\) (b) \(\frac{(25-18)^{2}}{18}+\frac{(25-22)^{2}}{22}+\frac{(25-39)^{2}}{39}+\frac{(25-21)^{2}}{21}\) (c) \(\frac{(18-25)}{25}+\frac{(22-25)}{25}+\frac{(39-25)}{25}+\frac{(21-25)}{25}\) (d) \(\frac{(18-25)^{2}}{100}+\frac{(22-25)^{2}}{100}+\frac{(39-25)^{2}}{100}+\frac{(21-25)^{2}}{100}\) (e) \(\frac{(0.18-0.25)^{2}}{0.25}+\frac{(0.22-0.25)^{2}}{0.25}+\frac{(0.39-0.25)^{2}}{0.25}\) \(+\frac{(0.21-0.25)^{2}}{0.25}\) The chi-square statistic is

Step by step solution

01

Understanding Expected Frequencies

First, calculate the expected frequency if the food preferences were evenly distributed. Since there are 4 types of food, each type is expected to have \(\frac{100}{4} = 25\) students favoring it.

02

Verify Calculation of Chi-Square Values

The Chi-square test formula utilized is \(\sum \frac{(O-E)^{2}}{E}\), where \(O\) is the observed frequency and \(E\) is the expected frequency. Match this to the options given.

03

Match with Corresponding Multiple Choice Option

A) uses the right formula and components: \(\frac{(18-25)^{2}}{25}+\frac{(22-25)^{2}}{25}+\frac{(39-25)^{2}}{25}+\frac{(21-25)^{2}}{25}\). This is the chi-square test formula applied correctly with expected frequencies of 25.

04

Eliminate Incorrect Options

Options B, C, D, and E use incorrect formulas or denominators that don't match the chi-square formula using observed and expected frequencies correctly.

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Key Concepts

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

Observed Frequencies

In statistical tests, observed frequencies refer to the actual number of occurrences per category that are collected during an experiment or survey. In this cafeteria food choice problem, the observed frequencies are the counts of students who prefer each type of food. For instance, the counts are: 18 for Asian food, 22 for Mexican food, 39 for Pizza, and 21 for Hamburgers. These observed frequencies are collected through direct questioning of a sample group and reflect their preferences. Understanding observed frequencies is crucial because these are the starting point in a chi-square test, acting as the ‘O’ in the chi-square formula \( \sum \frac{(O-E)^2}{E} \). In this formula, O represents the observed frequencies. Observed frequencies help highlight any potential differences in preferences among the different types of food offered.

Expected Frequencies

Expected frequencies are a theoretical count of occurrences that you would expect to see if there were no special influences affecting the categories being measured. In this context, it refers to the assumption that each food category is equally popular among students. To calculate the expected frequencies, you take the total number of observations (in this case, 100 students) and divide it by the number of categories, which are 4 types of food here. Thus, the expected frequency for each category is calculated as \( \frac{100}{4} = 25 \). Expected frequencies form the ‘E’ in our chi-square equation \( \sum \frac{(O-E)^2}{E} \), serving as the benchmark expected under random distribution. By comparing observed frequencies to expected frequencies, the chi-square test can determine if actual preferences deviate significantly from what was expected.

Statistical Analysis

Statistical analysis involves collecting, exploring, and interpreting large amounts of data. In the chi-square test, this means systematically comparing observed data with what would be expected under a null hypothesis. With our cafeteria example, statistical analysis is executed via the chi-square test to see if there's a significant difference between how students actually choose their meals and how they would if each type of food were equally preferred. The formula used is \( \chi^2 = \sum \frac{(O-E)^2}{E} \), where you sum these calculations for all food categories. This test is particularly useful in determining whether the observed variations in food preferences are statistically significant or merely due to random chance. Successful statistical analysis in this context will guide informed decisions regarding cafeteria supply and menu planning.

Cafeteria Food Choices

Understanding cafeteria food choices is essential for effectively meeting student dietary needs and preferences. By conducting surveys and analyzing food preferences, cafeteria managers can make data-driven decisions for ordering and menu planning. In this exercise, the survey covers preferences among Asian food, Mexican food, Pizza, and Hamburgers. By assessing if each food type holds equal popularity, stakeholders can manage inventory better, ensuring a balanced provision that aligns with actual demands. This not only aids logistical efficiency but also enhances student satisfaction by providing beloved meal options. Moreover, understanding these choices through a chi-square test helps track trending food preferences, which can change over time, aiding in both short-term and long-term nutritional planning.