Question

Is a customer's preference for a fast-food chain affected by the age of the customer? If so, advertising might need to target a particular age group. Suppose a random sample of 500 fast-food customers aged 16 and older was selected, and their favorite fast-food restaurants along with their age groups were recorded, as shown in the table: Use an appropriate method to determine whether or not a customer's fast-food preference is dependent on age. Write a short paragraph presenting your statistical conclusions and their practical implications for marketing experts.

Short answer

Answer: To determine whether there is a significant relationship between a customer's age and their fast-food preference, based on the chi-square test of independence, we need to follow these steps: 1. Create a contingency table using the given data. 2. Calculate the expected frequencies for each age group and restaurant combination. 3. Calculate the chi-squared test statistic. 4. Determine the degrees of freedom for the chi-square test. 5. Determine the critical value and p-value. 6. Draw conclusions and provide practical implications. Based on the results of the chi-square test (whether we reject or fail to reject the null hypothesis), we can conclude whether there is a significant relationship between a customer's age and their fast-food preference. If we reject the null hypothesis, this means that a customer's fast-food preference depends on their age, and marketing experts should consider targeting advertisement efforts towards specific age groups. If we fail to reject the null hypothesis, this means that a customer's fast-food preference is independent of their age, and marketing experts may not need to focus on the age factor when designing advertisement campaigns.

Step-by-step solution

Create a contingency table using the given data

By organizing the given data into a contingency table, we can see the relationship between age groups and fast-food preference. The rows represent the age groups and the columns represent the fast-food restaurants. Fill the cells with the corresponding frequency of each combination of age group and fast-food preference.

Calculate the expected frequencies for each age group and restaurant combination

The expected frequency for each cell is defined as the product of the corresponding row-total and column-total divided by the grand total. We can denote it as E_{ij} = (Row_i * Col_j) / N, where E_{ij} is the expected frequency for the cell at the intersection of row i and column j, Row_i is the total for i-th row, Col_j is the total for j-th column, and N is the grand total (sum of all row-totals or column-totals).

Calculate the chi-squared test statistic

To perform the chi-square test, we first need to calculate the chi-squared test statistic, denoted as X^2. It is the sum of the squared differences between the observed frequencies (O_{ij}) and expected frequencies (E_{ij}) divided by the expected frequencies, for all cells in the contingency table: X^2 = Σ((O_{ij} - E_{ij})^2 / E_{ij})

Determine the degrees of freedom for the chi-square test

The degrees of freedom (df) for a chi-square test of independence is calculated by multiplying the number of rows minus one (R-1) by the number of columns minus one (C-1): df = (R-1) * (C-1)

Determine the critical value and p-value

Using a chi-square distribution table or an online calculator, compare the test statistic (X^2) to the critical value corresponding to the chosen α-level (commonly α = 0.05) and the previously calculated df. Alternatively, calculate the associated p-value and compare it to the α-level. If the test statistic is greater than the critical value or the p-value is less than α, we reject the null hypothesis (the fast-food preference is independent of age).

Draw conclusions and provide practical implications

Based on the results of the chi-square test (whether we reject or fail to reject the null hypothesis), we can conclude whether there is a significant relationship between a customer's age and their fast-food preference. If we reject the null hypothesis, this means that a customer's fast-food preference depends on their age, and marketing experts should consider targeting advertisement efforts towards specific age groups. If we fail to reject the null hypothesis, this means that a customer's fast-food preference is independent of their age, and marketing experts may not need to focus on the age factor when designing advertisement campaigns.

Key concepts

Contingency Table

A contingency table is a simple yet powerful tool used in statistics to explore the relationship between two categorical variables. In the context of our exercise, the table helps illustrate how different age groups prefer different fast-food chains.
This type of table organizes the data into rows and columns, with each cell representing the frequency of occurrences for a specific combination of the two variables.

• Rows in the table represent one variable, such as different age groups in our study.
• Columns represent the other variable, such as the fast-food restaurant preferences.
• The cells contain the actual frequency counts or observations of that pair, denoting, for example, how many people of a certain age group prefer a specific fast-food chain.

To visualize the association between age and fast-food preferences, the contingency table summarizes data so underlying patterns or inconsistencies can be identified. This visual representation is crucial before proceeding with statistical analysis, as it offers a preliminary look at potential dependencies.

Degrees of Freedom

The concept of degrees of freedom (df) is central to understanding the chi-square test of independence. This term refers to the number of values in a calculation that are free to vary while calculating statistical estimates.
In a contingency table, degrees of freedom help determine the appropriate chi-square distribution to use when assessing independence.
The formula to calculate the degrees of freedom for a chi-square test of independence is:

• df = (R-1) * (C-1)

Here, R stands for the number of rows and C stands for the number of columns in the table.
For instance, if a table has 3 age groups and 4 different fast-food chains, then:

• df = (3-1) * (4-1) = 2 * 3 = 6

Degrees of freedom are critical because they dictate the shape and scale of the chi-square distribution used, which ultimately influences the hypothesis test's outcome.

Hypothesis Testing

Hypothesis testing is a methodology used to make inferences or claims about a population based on sample data. In this scenario, it helps determine whether a customer's age affects their fast-food preferences.
The process begins by developing two opposing hypotheses:

• Null Hypothesis (H₀): Assumes there is no relationship between age group and fast-food preference. Any observed differences are due to random chance.
• Alternative Hypothesis (H₁): Proposes that there is a significant association between age group and fast-food preference.

A chi-square test is then conducted on the contingency table data, producing a test statistic that helps decide whether to reject H₀. If the test indicates that the data significantly deviate from what H₀ suggests, we may reject the null hypothesis, supporting the alternative hypothesis and signaling a dependency between variables.

Critical Value

The critical value in a chi-square test of independence marks the cutoff point that determines when to reject the null hypothesis. It is based on a predetermined significance level (commonly \( \alpha = 0.05 \)) and the degrees of freedom computed from the contingency table.
The critical value serves as a benchmark for the test statistic, which is derived from the observed and expected frequencies in the table.

• If the chi-square test statistic exceeds the critical value, the null hypothesis (H₀) is rejected.
• If the statistic falls below this threshold, we fail to reject the null hypothesis.

This assessment is crucial for determining the existence of a significant relationship. Alternatively, calculating the p-value, which represents the probability of observing such data if the null hypothesis is true, can also lead to the same conclusion when compared to \( \alpha \).
Understanding the critical value enables researchers to control how often they risk rejecting a true null hypothesis, maintaining the integrity of their statistical inferences.