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Chapter 4: Problem 27

Exercise 4.26 discusses a sample of households in the US. We are interested in determining whether or not there is a linear relationship between household income and number of children. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Which sample correlation shows more evidence of a relationship, \(r=0.25\) or \(r=0.75 ?\) (c) Which sample correlation shows more evidence of a relationship, \(r=0.50\) or \(r=-0.50 ?\)

Short Answer

Expert verified
Parameters: household income and number of children; Null hypothesis: \(H_0\): r = 0, Alternative hypothesis: \(H_A\): r 鈮� 0. \(r=0.75\) shows stronger evidence of a relationship than \(r=0.25\). Both \(r=0.50\) and \(r=-0.50\) show the same strength of evidence but in opposite directions.

Step by step solution

01

Define Parameters

The relevant parameters in this case are the household income and the number of children. The null hypothesis (\(H_0\)) would state that there is no linear relationship between the two, i.e., the correlation coefficient r = 0. The alternative hypothesis (\(H_A\)) would suggest that there is a significant linear relationship, i.e., r 鈮� 0.
02

Analyze Sample Correlation

A correlation of r = 0.75 shows more evidence of a relationship between household income and number of children compared to r = 0.25. The sign indicates the direction of the relationship and not the strength, thus only the magnitude of r is considered for strength of evidence.
03

Compare Positive and Negative Correlations

Both r = 0.50 and r = -0.50 provide the same strength of evidence of a relationship as they both have the same magnitude but in opposite directions. A positive correlation (r = 0.50) indicates that as household income increases, so does the number of children, and a negative correlation (r = -0.50) suggests that as household income increases, the number of children decreases.

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Key Concepts

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

Linear Relationship
A linear relationship describes a straight-line connection between two variables. In the given exercise, the variables are household income and the number of children within a household. This type of relationship is significant because it allows us to predict one variable based on the value of the other. For example, if we find a positive linear relationship, this means that as household income increases, the number of children tends to increase as well. Conversely, a negative linear relationship would imply that as one increases, the other decreases.

A key component to identifying linear relationships is the correlation coefficient (r), which ranges from -1 to 1. A correlation closer to 1 indicates a strong positive linear relationship, closer to -1 a strong negative linear relationship, and close to 0 indicates no linear relationship. In practical terms, this coefficient helps quantify the degree of the relationship and gives us a mathematical approach to understanding the interaction between the two variables.
Null Hypothesis
The null hypothesis ( H_0 ) is a foundational concept in hypothesis testing. It is a statement suggesting that there is no significant effect or relationship between two variables. In correlation analysis, the null hypothesis typically posits the absence of a linear relationship. In this exercise, the null hypothesis states that there is no linear relationship between household income and the number of children.
  • This means that any observed correlation in the sample data is due merely to random chance.
  • The null hypothesis is assumed to be true until evidence suggests otherwise.
It acts as a starting point for statistical testing, allowing researchers to use sample data to assess the possibility of rejecting this hypothesis in favor of an alternative one. By testing the null hypothesis, we can determine if the sample correlation coefficient is significantly different from zero, indicating a possible linear relationship exists.
Alternative Hypothesis
The alternative hypothesis ( H_A ) is a key part of hypothesis testing, offering a contrast to the null hypothesis. In correlation studies, the alternative hypothesis indicates that a linear relationship does indeed exist between the pair of variables. For the given problem, the alternative hypothesis suggests that there is a significant linear relationship between household income and the number of children. This means that any observed correlation is statistically significant and not just due to random variation in the sample data.
  • If evidence supporting the alternative hypothesis is strong, the null hypothesis is rejected.
  • This decision process involves examining the correlation coefficient and deciding if the strength of the relationship (either positive or negative) is sufficient to challenge the null hypothesis.
By accepting the alternative hypothesis, we conclude that changes in one variable are associated with changes in the other, consequently allowing us to understand more about the dynamics between household income and family size.

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Most popular questions from this chapter

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