Sport Preference Poll Results For Men

A sport preference poll yielded the following data for men and women. Use a 5% significance level and test to determine if sport preference and gender are independent.

A sport preference poll collected data on the preferences for basketball, football, and soccer among men and women. The task is to examine whether sport preference is dependent on gender. To do this, we will apply a chi-square test of independence at a significance level of 5%. The analysis involves calculating the test statistic based on the observed and expected frequencies, determining the critical value from the chi-square distribution table, and drawing a conclusion regarding the independence of sport preference and gender.

The chi-square test of independence is an essential statistical method used to examine whether two categorical variables are related or independent. In this context, the variables are gender (men and women) and sport preference (basketball, football, soccer). If the test indicates a significant association, it suggests that sport preference depends on gender; otherwise, the two variables are considered independent.

First, tabulated data must be organized with observed frequencies for each sport preference within gender categories. Based on the data, the expected frequencies are computed under the null hypothesis of independence, which posits that sport preference and gender are unrelated. The chi-square test statistic is then calculated by summing the squared differences between observed and expected frequencies divided by the expected frequencies across all categories.

The test value, also known as the chi-square statistic (χ²), quantifies the discrepancy between observed and expected frequencies. To determine the test value, we sum all the components: (Observed - Expected)² / Expected. The critical value is obtained from the chi-square distribution table with degrees of freedom equal to (number of rows - 1) multiplied by (number of columns - 1), which accounts for the categories being tested.

If the computed chi-square statistic exceeds the critical value at the 5% significance level, we reject the null hypothesis of independence, concluding that sport preference and gender are associated. Conversely, if the statistic falls below the critical value, we fail to reject the null hypothesis, indicating insufficient evidence to claim dependence.

Based on the given options:

1. There is sufficient evidence to support the claim that one's sport preference is dependent on one's gender.

2. There is not sufficient evidence to support the claim that one's sport preference is dependent on one's gender.

The analysis's outcome depends on the calculated test statistic and the critical value. A detailed numerical solution involving data is necessary to precisely compute these values. However, the general approach involves setting up the contingency table, calculating row and column totals, expected frequencies, and the chi-square statistic. If the specific data were provided, this process would yield a definitive conclusion.

In conclusion, the chi-square test of independence is a vital statistical tool for analyzing categorical data. When applied appropriately, it helps determine whether two variables, such as sport preference and gender, are related or independent. Such analyses aid in understanding behavioral patterns and preferences, which can inform marketing strategies, resource allocation, and further research in sports sociology.

Paper For Above instruction

The relationship between gender and sport preference has long been of interest to sports marketers, sociologists, and sports organizations. Understanding whether sport preference depends on gender can influence marketing strategies, the structuring of sports leagues, and community engagement initiatives. This paper explores the application of the chi-square test of independence to determine if sport preferences are associated with gender, based on a hypothetical dataset collected from a sport preference poll.

Firstly, it is necessary to understand the principles behind the chi-square test of independence. The test is employed when examining the relationship between two categorical variables, in this case, gender (male and female) and sport preference (basketball, football, soccer). The null hypothesis (H0) claims that there is no association between the variables; they are independent. The alternative hypothesis (H1) posits that the variables are dependent, indicating some relationship between gender and sport preference.

Data organization is the first step in conducting the test. The observed frequencies are tabulated in a contingency table, showing counts of men and women preferring each sport. For analytical purposes, the total counts across all categories are summed, providing marginal totals. This organized structure facilitates the calculation of expected frequencies under the assumption of independence, which are derived from the product of the corresponding row and column totals divided by the overall total.

Calculating the expected frequencies involves the formula:

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

Once these expected frequencies are computed, the chi-square statistic is calculated using the formula:

χ² = Σ (Observed - Expected)² / Expected

The summation covers all categories within the contingency table. This statistic measures the deviation of observed data from what we would expect if the variables were indeed independent. A larger χ² indicates a greater discrepancy, implying potential dependence.

Next, the critical value for the χ² distribution, at a 5% significance level, is determined based on the degrees of freedom:

Degrees of freedom = (Number of rows - 1) × (Number of columns - 1)

For instance, with two genders and three sport preferences, the degrees of freedom would be (2-1)×(3-1)=2. Consulting the chi-square distribution table provides the critical value corresponding to this degree of freedom at the 0.05 significance level.

Comparing the calculated chi-square value with the critical value yields the decision rule:

- If χ² > critical value, reject H0 and conclude that sport preference is dependent on gender.

- If χ²

Applying this to example data, suppose the computed chi-square statistic was 9.21, and the critical value at 2 degrees of freedom is 5.991. Since 9.21 > 5.991, we reject the null hypothesis and conclude that sport preference is dependent on gender.

The implications of such a finding are significant. If sport preferences are indeed dependent on gender, sports organizations might tailor marketing campaigns differently for men and women, develop gender-specific sports programs, and allocate resources accordingly to optimize engagement and participation.

In summary, the chi-square test of independence provides a robust statistical framework for exploring the relationship between categorical variables such as gender and sport preference. Proper application of this test involves organizing data, computing expected frequencies, evaluating the chi-square statistic, and comparing it to the critical value to draw meaningful conclusions. Understanding these relationships enriches our appreciation of sports sociology and helps inform strategies for promoting sports participation across different demographic groups.

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