Question

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

Parameters: household income and number of children; Null hypothesis: \(H_0\): r = 0, Alternative hypothesis: \(H_A\): r ≠ 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

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 ≠ 0.

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.

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.