91Ó°ÊÓ

The paper "Credit Card Misuse, Money Attitudes, and Compulsive Buying Behavior: Comparison of Internal and External Locus of Control Consumers" (College Student Journal [2009]: \(268-275\) ) describes a study that surveyed a sample of college students at two midwestern public universities. Based on the survey responses, students were classified into two "locus of control" groups (internal and external) based on the extent to which they believe that they control what happens to them. Those in the internal locus of control group believe that they are usually in control of what happens to them, whereas those in the external locus of control group believe that it is usually factors outside their control that determines what happens to them. Each student was also classified according to a measure of compulsive buying. The resulting data are summarized in the accompanying table. Can the researchers conclude that there are an association between locus of control and compulsive buying behavior? Carry out a \(X^{2}\) test using \(\alpha=.01\). Assume it is reasonable to regard the sample as representative of college students at midwestern public universities. $$ \begin{array}{l|ccc} & & \text { Locus of Control } \\ \hline & \text { Internal } & \text { External } \\ \hline \text { Compulsive } & \text { Yes } & 3 & 14 \\ \text { Buyer? } & \text { No } & 52 & 57 \\ \hline \end{array} $$

Step by step solution

01

Set up hypotheses

The null hypothesis (\(H_{0}\)) assumes there is no association between the two categorical variables, locus of control and compulsive buying. The alternative hypothesis (\(H_{A}\)) assumes there is an association. \n\n\(H_{0}\): locus of control and compulsive buying are independent. \n\n\(H_{A}\): locus of control and compulsive buying are not independent.

02

Calculate the Expected Frequencies

Under the null hypothesis, we assume the variables are independent. This leads to an expected frequency for each cell in the table that can be calculated using the formula \(\frac{(RowTotal * ColumnTotal)}{GrandTotal}\).Expected frequencies: For Compulsive Buyers with Internal locus of control: \(\frac{(3+14)*(3+52)}{126} = 14.571\)For Compulsive Buyers with External locus of control: \(\frac{(3+14)*(14+57)}{126} = 2.429\)For Non-Compulsive Buyers with Internal locus of control: \(\frac{(52+57)*(3+52)}{126} = 40.429\)For Non-Compulsive Buyers with External locus of control: \(\frac{(52+57)*(14+57)}{126} = 68.571\

03

Calculate the Chi-Square Statistic

The chi-square statistic measures the divergence of the observed frequencies from the expected frequencies. It is calculated using the formula: \(\sum \frac{(Observed-Expected)^{2}}{Expected}\). Calculation:For Compulsive Buyers with Internal locus of control: \(\frac{(3-14.571)^2}{14.571} = 9.231\)For Compulsive Buyers with External locus of control: \(\frac{(14-2.429)^2}{2.429} = 55.368\)For Non-Compulsive Buyers with Internal locus of control: \(\frac{(52-40.429)^2}{40.429} = 3.310\)For Non-Compulsive Buyers with External locus of control: \(\frac{(57-68.571)^2}{68.571} = 1.945\)Sum these calculated values to get the chi-square statistic: \(9.231 + 55.368 + 3.310 + 1.945 = 69.854\

04

Find the critical value and make the decision

The degrees of freedom (df) can be calculated as (R-1)*(C-1) where R is the number of rows and C is the number of columns. In this case, df = (2 - 1)*(2 - 1) = 1. Looking at the chi-square distribution table with \(\alpha = .01\) and df=1, the critical chi-square value is 6.635.Comparing the calculated chi-square statistic with the critical value, 69.854 > 6.635, we reject the null hypothesis. This means there is significant evidence to say the locus of control and compulsive buying are not independent for the students.

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Key Concepts

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

Null Hypothesis

The null hypothesis, symbolized as \(H_0\), is a fundamental concept in statistics that refers to a general statement or default position that there is no relationship between two measured phenomena. In our case, it implies that locus of control and compulsive buying behavior are independent of each other—that the belief in personal control over one's life does not affect one’s shopping habits.

Understanding the null hypothesis provides a basis for statistical testing. In a Chi-Square test of independence, like the one in the exercise, we start by presuming the null hypothesis is true. Then, we use sample data to determine if there's enough evidence to reject this assumption in favor of an alternative hypothesis, which suggests that there is, in fact, a relationship between the variables being studied.

Expected Frequencies

Expected frequencies are the counts we would anticipate in each category of a contingency table if the null hypothesis were true. They're crucial because the Chi-Square test compares these expected counts to the actual observed counts to see if there's a significant discrepancy.

To calculate expected frequencies, you use the formula \(\frac{(RowTotal \times ColumnTotal)}{GrandTotal}\), ensuring that the row and column sums match those in the actual data. This process assumes that the presence or lack of one characteristic is independent of the presence or lack of another, which is the essence of the null hypothesis in a test of independence.

Locus of Control

Locus of control is a psychological concept that pertains to the degree to which individuals believe they have control over the events in their lives. Those with an internal locus of control believe that their own actions and decisions directly impact outcomes, whereas individuals with an external locus of control feel that external forces, such as luck or fate, play a larger role.

Understanding one's locus of control can be important in predicting behavior, such as compulsive buying. The exercise examines whether a person's perceived control over their life (internal vs. external locus of control) correlates with their likelihood to engage in compulsive buying. This psychological principle can offer insights into consumer behavior and financial decision-making among different groups of people.

Compulsive Buying Behavior

Compulsive buying behavior is a pattern of chronic, repetitive purchasing that becomes a primary response to negative events or feelings. It's often associated with a loss of control, significant distress, and functional impairment. Understanding compulsive buying is important for studying consumer habits and financial management.

The behavior is relevant to the exercise because the study aims to see if there's a link between locus of control and the tendency towards compulsive buying. Recognizing such patterns can help in designing interventions, informing educational programs, and assisting individuals in developing healthier buying habits.

Statistical Significance

Statistical significance is a determination that an observed effect in data is not likely due to chance alone. It provides a quantitative measure for evaluating the null hypothesis. The significance level, denoted as \(\alpha\), is a threshold for deciding whether to reject the null hypothesis; commonly used \(\alpha\) levels are 0.05 or 0.01.

In the context of our exercise, we determine statistical significance by comparing the calculated Chi-Square statistic to a critical value from the Chi-Square distribution. If the Chi-Square value exceeds the critical value at the chosen \(\alpha\) level, we consider the result statistically significant. This implies that the findings are unlikely to have occurred by chance, suggesting a relationship between locus of control and compulsive buying behavior.