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Chapter 4: Problem 27

Relationship between Income and Number of Children 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
The null hypothesis is that the correlation coefficient between household income and number of children is zero, alternative hypothesis is that it's not zero. When comparing \(r=0.25\) and \(r=0.75\), \(r=0.75\) shows stronger relationship, while when comparing \(r=0.50\) and \(r=-0.50\), both show the same strength of a relationship but in opposite directions.

Step by step solution

01

Define the relevant parameter(s) and state the null and alternative hypotheses.

The relevant parameter here is the correlation coefficient (\(r\)) between household income and number of children in a family. The null hypothesis (\(H_0\)) is that there is no relationship, i.e., \(r = 0\), and the alternative hypothesis (\(H_1\)) is that there is a relationship, i.e., \(r \neq 0\) in the population.
02

Compare the evidence of the relationship for \(r=0.25\) and \(r=0.75\)

The absolute value of the correlation coefficient measures the strength and direction of a linear relationship between two variables. A coefficient closer to -1 or 1 indicates a stronger relationship. \(r=0.75\) is closer to 1 hence it indicates a stronger relationship between household income and number of children when compared to \(r=0.25\). Therefore, \(r=0.75\) shows more evidence of a relationship.
03

Compare the evidence of the relationship for \(r=0.50\) and \(r=-0.50\)

The sign of the correlation coefficient (\(r\)) determines the direction of the relationship, with negative indicating inverse relationship, and positive indicating direct relationship. However, the strength of the relationship is judged by absolute values. Thus, both \(r=0.50\) and \(r=-0.50\) show the same strength of evidence about a linear relationship between household income and number of children (though in opposite directions).

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Key Concepts

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

Linear Relationship
When we speak of a linear relationship in statistics, we refer to a situation where if one variable increases, the other variable tends to increase as well, if the relationship is positive, or decrease if the relationship is negative. This is visually represented by a straight line when plotted on a graph. In the context of household income and number of children, if we say there is a linear relationship, we are suggesting that changes in the number of children a household has could potentially be associated with changes in the household鈥檚 income, either positively or negatively.
Null and Alternative Hypotheses
When researchers set out to determine if there is a relationship between two variables, null and alternative hypotheses are formulated. The null hypothesis, symbolized as H0, typically suggests that no effect or no difference will be found. In this case, the null hypothesis posits that there is no linear relationship between household income and number of children. On the contrary, the alternative hypothesis, H1, represents what researchers expect to find; that is, there is a linear relationship between household income and the number of children. These hypotheses lay the groundwork for statistical testing.
Household Income
Household income is a critical measure used in many sociological and economic studies as it reflects the total earnings of all members living in a single household. This measure is important in this context because it provides insight into the economic status of a family, which may influence and relate to many other factors, such as the number of children a family may decide to have or can financially support.
Number of Children
The variable number of children in a household is a straightforward count of how many children are living within a family unit. It's a demographic variable that often interacts with various socio-economic factors, including household income. The relationship explored here aims to identify if these counts can influence, or be influenced by, the economic standing represented by the household鈥檚 income.
Statistical Evidence
Statistical evidence pertains to the proof or data that support or refute a hypothesis, based on statistical analysis. When comparing two sample correlation coefficients, such as r=0.25 and r=0.75, the one with greater absolute value (r=0.75) provides stronger statistical evidence supporting the alternative hypothesis. This implies a more significant relationship between the variables, hence stronger evidence in context to our interest in the relationship between household income and the number of children.
Strength of Relationship
The strength of relationship in a statistical context is about how well the relationship between two variables can be described using a straight line. Correlation coefficients can range from -1 to 1. The closer the value is to -1 or 1, the stronger the evidence of a relationship. A value of 0 means there is no linear relationship. In our case, r=0.75 indicates a stronger relationship compared to r=0.25. It is essential to note that both r=-0.50 and r=0.50 represent relationships of equal strength but in opposite directions, emphasizing that while the direction of the relationship (positive or negative) is relevant, the absolute value determines the strength.

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Most popular questions from this chapter

Two p-values are given. Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.04\) or \(\quad\) p-value \(=0.62\)

Significant and Insignificant Results (a) If we are conducting a statistical test and determine that our sample shows significant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (b) If we are conducting a statistical test and determine that our sample shows insignificant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (c) Explain why we generally won't ever know which of the realities (in either case) is correct.

Classroom Games: Is One Question Harder? Exercise 4.62 describes an experiment involving playing games in class. One concern in the experiment is that the exam question related to Game 1 might be a lot easier or harder than the question for Game \(2 .\) In fact, when they compared the mean performance of all students on Question 1 to Question 2 (using a two-tailed test for a difference in means), they report a p-value equal to 0.0012 . (a) If you were to repeat this experiment 1000 times, and there really is no difference in the difficulty of the questions, how often would you expect the means to be as different as observed in the actual study? (b) Do you think this p-value indicates that there is a difference in the average difficulty of the two questions? Why or why not? (c) Based on the information given, can you tell which (if either) of the two questions is easier?

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \mu_{1}=\mu_{2} \operatorname{vs} H_{a}: \mu_{1}>\mu_{2} $$

Mice and Pain Can you tell if a mouse is in pain by looking at its facial expression? A new study believes you can. The study \(^{12}\) created a "mouse grimace scale" and tested to see if there was a positive correlation between scores on that scale and the degree and duration of pain (based on injections of a weak and mildly painful solution). The study's authors believe that if the scale applies to other mammals as well, it could help veterinarians test how well painkillers and other medications work in animals. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Since the study authors report that you can tell if a mouse is in pain by looking at its facial expression, do you think the data were found to be statistically significant? Explain. (c) If another study were conducted testing the correlation between scores on the "mouse grimace scale" and a placebo (non-painful) solution, should we expect to see a sample correlation as extreme as that found in the original study? Explain. (For simplicity, assume we use a placebo that has no effect on the facial expressions of mice. Of course, in real life, you can never automatically assume that a placebo has no effect!) (d) How would your answer to part (c) change if the original study results showed no evidence of a relationship between mouse grimaces and pain?
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