91Ó°ÊÓ

Where do young adults live? A survey by the National Institutes of Health asked a random sample of young adults (aged 19 to 25 years), "Where do you live now? That is, where do you stay most often?" Here is the full two-way table (omitting a few who refused to answer and one who claimed to be homeless):" $$ \begin{array}{lcc} \hline & \text { Female } & \text { Male } \\ \text { Parents' home } & 923 & 986 \\ \text { Another person's home } & 144 & 132 \\ \text { Own place } & 1294 & 1129 \\ \text { Group quarters } & 127 & 119 \\ \hline \end{array} $$ (a) Should we use a chi-square test for homogeneity or a chi-square test for independence in this setting? Justify your answer. (b) State appropriate hypotheses for performing the type of test you chose in part (a). Minitab output from a chi-square test is shown below. (c) Check that the conditions for carrying out the test are met. (d) Interpret the \(P\) -value in context. What conclusion would you draw?

Step by step solution

01

Determine the Type of Test

In this scenario, we are given data from a two-way table summarizing the living arrangements of young adults by gender. A chi-square test for independence determines if two categorical variables are independent. The two variables here are 'gender' and 'location of residence'. Since we have one sample and two categorical variables, the chi-square test for independence is appropriate.

02

State the Hypotheses

For a chi-square test for independence, the null hypothesis ( H_0 ) is that there is no association between the two categorical variables, so 'gender' and 'location of residence' are independent. The alternative hypothesis ( H_a ) is that there is an association between 'gender' and 'location of residence'.

03

Check Test Conditions

Check if all expected counts are at least 5, which is a condition for using the chi-square test. Calculate the expected counts for each cell using the formula: \[ \text{Expected Count} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}} \] If all expected counts meet this condition, the chi-square test can be performed.

04

Interpret the P-value

The P-value given in the Minitab output represents the probability that the observed data (or more extreme) is due to chance, assuming the null hypothesis is true. If the P-value is less than the significance level (typically 0.05), we reject the null hypothesis. A low P-value indicates that there is likely an association between gender and location of residence.

05

Draw a Conclusion

Based on the P-value, if it is below the significance level, we conclude that there is an association between gender and living arrangements among young adults. If it is above the threshold, we do not have enough evidence to suggest an association and thus fail to reject the null hypothesis.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Two-way table analysis

A two-way table is a fantastic tool to help organize and understand data when dealing with two categorical variables. In this exercise, the two-way table provides a clear overview of where young adults live based on their gender. This layout allows for easy comparison of the living arrangements between males and females.

Two-way tables help summarize large amounts of data neatly. They reveal patterns and relationships that might not be immediately obvious. Each cell in the table represents a count of individuals sharing the same characteristics in terms of both categories (e.g., males living at parents' home).

Analyzing these tables can provide insights into whether there might be an association between the two variables—"gender" and "where young adults live" in this case. This setup is ideal for conducting statistical tests, such as the chi-square test, to further explore potential relationships.

Gender differences

Gender differences refer to the distinctions in behaviors, preferences, and opportunities attributed to males and females. In statistical studies like this survey, exploring how gender influences living arrangements can uncover significant trends and patterns.

By examining the data, we can assess whether males and females tend to live in similar or different types of residences. Are males more inclined towards living on their own, or perhaps women prefer staying in their own place? These are a few questions explored through the chi-square test for independence, which assesses if gender and living arrangement are related.

Understanding these differences can lead to valuable insights into social norms or economic factors influencing each gender's living conditions. Insights drawn from such studies can help in developing policies or targeted measures to address any detected disparities.

Living arrangements

Living arrangements provide an essential glimpse into the lifestyle and socio-economic condition of individuals, especially young adults in this case. The survey examines where young adults typically reside—whether it’s their parents' homes, their place, group quarters, or someone else's home.

Different living arrangements can signify different life stages or circumstances; for instance, living at parents' home might be more common among students or those starting their careers. Meanwhile, having one's own place might indicate financial independence.

This diversity in living arrangements helps identify patterns across genders, showing whether cultural or economic factors play a significant role in where young individuals choose to reside. By understanding these patterns, we can better cater to the needs and challenges faced by different groups.

Statistical independence

Statistical independence is a crucial concept in understanding relationships between variables. When two variables are statistically independent, the distribution of one variable is not affected by the other. In simpler terms, knowing the value of one variable doesn't tell you anything about the other.

In the context of this exercise, we use a chi-square test for independence to determine if gender and living arrangement are independent: whether knowing a person's gender gives us information on their likely living arrangement.

If our test shows these variables are not independent (meaning there is an association), it suggests that gender influences where individuals live. This information can be vital for sociological studies, highlighting areas for further research or policy intervention if a meaningful association is found.