Question

The "sandwich generation" refers to middle-aged Americans who are either providing support for an aging parent while raising child under 18 or supporting a child over 18 . The information that follows summarizes the results of a poll \(^{8}\) regarding support for a parent 65 years or older by demographic group. $$ \begin{array}{lrrr} \hline \text { Provide Financial } & & & \\ \text { Support } & \text { Yes } & \text { No } & \text { Total } \\ \hline \text { Hispanics } & 134 & 66 & 200 \\ \text { Blacks } & 86 & 114 & 200 \\ \text { Whites } & 49 & 151 & 200 \\ \hline \end{array} $$ a. Find the proportion of individuals providing financial support for their parents for each demographic group. b. Are there significant differences among the proportions providing financial support for their parents for these demographic groups of Americans? Use \(\alpha=.01\). Are your results consistent with the observed proportions in part a?

Short answer

** Answer: To answer this question, first, calculate the proportions of individuals providing financial support for their parents in each demographic group: Hispanics: 0.67 Blacks: 0.43 Whites: 0.245 Next, perform a hypothesis test using a Chi-Square test of independence with a level of significance, α=0.01. The null hypothesis (H0) is that there is no difference among the proportions, while the alternative hypothesis (Ha) is that there is a difference. After calculating the Chi-Square test statistic and comparing it with the critical value, we can either reject or fail to reject the null hypothesis. If the test statistic is higher than the critical value, we reject the null hypothesis, which means there is a significant difference among the proportions providing financial support for their parents for these demographic groups. If the test statistic is not higher than the critical value, we fail to reject the null hypothesis, which means there is not enough evidence to support the claim of significant differences among the proportions. Based on the specified steps and interpretation of the test results, we can conclude if there are significant differences among the proportions providing financial support for their parents for these demographic groups and if the results are consistent with the observed proportions (0.67 for Hispanics, 0.43 for Blacks, and 0.245 for Whites).

Step-by-step solution

Calculate proportions for each demographic group.

To find the proportion of individuals providing financial support for their parents in each demographic group, divide the number of individuals providing support by the total number of individuals in the group: Hispanics: \(\dfrac{134}{200} = 0.67\) Blacks: \(\dfrac{86}{200} = 0.43\) Whites: \(\dfrac{49}{200} = 0.245\)

Stating the null and alternative hypotheses.

We will perform a hypothesis test to determine if there are significant differences in the proportions among the three demographic groups. The null hypothesis (H0) is that there is no difference among the proportions, while the alternative hypothesis (Ha) is that there is a difference. H0: p1 = p2 = p3 Ha: At least one proportion is different.

Conduct the hypothesis test using chi-square test.

To conduct the hypothesis test, use the Chi-Square test of independence. Using a statistical software or calculator, enter the values from the table into a 2x3 contingency table: $$ \begin{array}{lrrr} \hline & \text{Yes} & \text{No} & \text{Total}\\ \hline \text{Hispanics} & 134 & 66 & 200\\ \text{Blacks} & 86 & 114 & 200\\ \text{Whites} & 49 & 151 & 200\\ \hline \end{array} $$ Make sure to use a level of significance, α=0.01. Calculate the Chi-Square test statistic (X-squared) and the critical value at the specified alpha level.

Interpret the test results and draw conclusions.

Compare the calculated Chi-Square test statistic with the critical value. If the test statistic is higher than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Based on the results, either reject or fail to reject the null hypothesis and conclude whether there are significant differences among the proportions providing financial support for their parents for these demographic groups. Assert if the results are consistent with the observed proportions calculated in part a.

Key concepts

Hypothesis Testing

In research and statistics, hypothesis testing is a critical method for making decisions using data. Whether comparing academic scores, medical treatment effectiveness, or, as in our exercise, financial support across demographic groups, hypothesis testing allows us to draw conclusions about populations based on sample data.

The process begins with the formulation of null (\(H_0\) and alternative (\(H_a\) hypotheses. The null hypothesis typically represents no effect or no difference, and serves as a default assumption to test against. In the context of the provided exercise, the null hypothesis suggests that the proportion of individuals who provide financial support to their elderly parents is the same across Hispanics, Blacks, and Whites.

Researchers then use statistical tests to determine if there is enough evidence to reject the null hypothesis in favor of the alternative, which suggests that there is a significant difference. The Chi-Square test of independence, used in our scenario, is particularly suited for categorical data to assess whether two variables are independent of each other.

Proportions

A proportion is a statistical measure that represents the size of one group relative to another. It is often expressed as a fraction or percentage. In demographics, this could be the ratio of a particular ethnic group within a community, or, as in our sandwich generation example, the portion of individuals providing financial support to an aging parent.

Calculating proportions is straightforward: divide the count of the subset by the total count. These values are crucial for various statistical analyses, including the Chi-Square test in part a of the exercise. Proportions provide a clear and quantifiable way to compare groups. In the exercise, understanding the separate proportions among Hispanics (\(0.67\)), Blacks (\(0.43\)), and Whites (\(0.245\)) offers a baseline for detecting if the differences observed are significant or not.

Statistical Significance

The term statistical significance is a cornerstone of hypothesis testing. It is a measure of how likely it is that an obtained result occurred by chance. The p-value, obtained from a statistical test, helps to determine significance. A result is typically considered statistically significant if the p-value is less than the chosen significance level, denoted as \( \alpha \).

In our example, \( \alpha = 0.01 \) sets a high standard for significance, meaning there is only a 1% chance we would conclude there is a difference when there is none (Type I error). Compare the p-value from the Chi-Square test to \( \alpha \). If it's lower, the data provide enough evidence to reject the null hypothesis, implying the differences in demographic proportions are unlikely due to random chance. This would suggest a meaningful discrepancy in the financial support patterns among the different demographic groups.

Demographic Analysis

In summary, demographic analysis involves the examination of data regarding the structure of populations. It often includes an analysis of variables such as age, race, or gender to understand patterns and trends within different groups.

In our poll regarding financial support for aging parents, demographic analysis allows us to dissect the data into meaningful segments and assess differences and disparities among these segments. Using demographic analysis in conjunction with the Chi-Square test, we can interpret not only the proportions of individuals providing financial support but also the cultural or socioeconomic factors that could influence these proportions. The insights gained can guide policies and programs tailored to the unique needs represented within each demographic group, and provide a more nuanced understanding of the societal phenomena described as the 'sandwich generation' dilemma.