Chi Square Analysis Worksheet - Mth 160 Statistics Version 1

Chi Square Analysis Worksheet Mth 160 Statisticsversion 1 0913sup

Determine whether there is a relationship between a student’s gender and a student’s major in college by analyzing survey data from 55 students across categories of Natural Science, Social Sciences, and Humanities. Conduct a chi-square test of independence, including calculation of expected frequencies, chi-square statistic, and interpretation of results. Additionally, compare the distribution of women’s majors in the sample to national percentages using a chi-square goodness-of-fit test. Follow all steps with detailed calculations and reasoning.

Paper For Above instruction

Introduction

The application of chi-square tests in statistical analysis provides vital insights into categorical data, especially in contexts examining relationships or distribution fitting. In this study, we explore two primary questions using chi-square methods: first, whether there is a relationship between gender and college major, and second, whether the distribution of women’s majors in our sample aligns with national percentages. These analyses contribute to understanding gender-based major choices and the representativeness of the sample.

Part I: Testing Independence Between Gender and College Major

1. Calculating Expected Frequencies

The contingency table categorizes 55 students by gender and major. To determine expected frequencies for each cell under the assumption of independence, we use the formula:

Expected frequency = (Row total * Column total) / Grand total

Suppose the observed data are as follows:

Using the above, expected frequencies for each cell are calculated. For example, expected students who are men and major in Natural Science:

ExpectedMen, NS = (OMen, Total * TNS) / 55

The same approach applies for all six cells, ensuring calculations reflect the proportions under the null hypothesis of independence.

2. Computing the Chi-square Statistic

The chi-square statistic is computed using:

χ² = Σ [(O - E)² / E]

where O is the observed frequency, and E is the expected frequency for each cell. This involves calculating the squared difference between observed and expected, divided by the expected, for all cells, then summing these values.

3. Conducting the Chi-square Test

With the chi-square statistic calculated, we compare it to the critical value from the chi-square distribution table with appropriate degrees of freedom:

df = (number of rows - 1)  (number of columns - 1) = (2 - 1)  (3 - 1) = 2

If the calculated χ² exceeds the critical value at our chosen significance level (e.g., α=0.05), we reject the null hypothesis, indicating a significant association between gender and major.

4. Interpretation of Results

If the test is significant, it suggests that gender and college major are related within the sample. A non-significant result would imply independence, indicating no evidence of a relationship between gender and the choice of major based on the data. Limitations such as sample size and potential biases should also be considered when interpreting these results.

Part II: Goodness-of-Fit Test for Women’s Majors Against National Percentages

1. Expected Frequencies Based on National Percentages

Given the percentages: 22% in Natural Sciences, 28% in Social Sciences, and 30% in Humanities, the expected frequencies for women in the sample are calculated as:

ENS = total women * 0.22

ESS = total women * 0.28

EH = total women * 0.30

These expected counts are compared to the observed counts within the sample to test whether the distribution of women’s majors aligns with national trends.

2. Chi-square Goodness-of-Fit Calculation

The test statistic is computed similarly:

χ² = Σ [(O - E)² / E]

where the summation includes the categories of majors. Degrees of freedom for this test are:

df = number of categories - 1 = 3 - 1 = 2

Comparing the calculated χ² to the critical value determines if the sample distribution fits the national distribution.

3. Results Interpretation

A significant result indicates the sample’s distribution of women across majors significantly differs from national percentages, suggesting demographic or institutional differences. A non-significant result suggests the sample is representative of the national distribution.

Conclusion

Applying chi-square analyses enables researchers to assess relationships between categorical variables and distributional fits effectively. In our case, the independence analysis may reveal gender influences on major choices, while the goodness-of-fit test evaluates the representativeness of the sample’s female majors relative to national data. These insights are vital for institutional planning, policy formulation, and understanding student preferences. Limitations inevitable in sampling and data collection should temper the conclusions, emphasizing the importance of large, random samples and comprehensive data collection for accuracy and generalizability.

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