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Do taller adults make more money? The authors of the paper "Stature and Status: Height, Ability, and Labor Market Outcomes" (Journal of Political Economics [2008] : \(499-532\) ) investigated the association between height and earnings. They used the simple linear regression model to describe the relationship between \(x=\) height (in inches) and \(y=\log (\) weekly gross earnings in dollars) in a very large sample of men. The logarithm of weekly gross earnings was used because this transformation resulted in a relationship that was approximately linear. The paper reported that the slope of the estimated regression line was \(b=0.023\) and the standard deviation of \(b\) was \(s_{b}=0.004\). Carry out a hypothesis test to decide if there is convincing evidence of a useful linear relationship between height and the logarithm of weekly earnings. Assume that the basic assumptions of the simple linear regression model are reasonably met.

Step by step solution

01

State the Hypotheses

The null hypothesis (\(H_{0}\)): There is no linear relationship between adults' height and the logarithm of their weekly earnings, which means that the slope of the regression line \(b = 0\). The alternative hypothesis (\(H_{1}\)): There is a significant linear relationship between adults' height and the logarithm of their weekly earnings, meaning that the slope of the regression line \(b\neq 0\).

02

Calculate Test Statistic

The test statistic for this hypothesis test is calculated as follows: \(t = \frac{b - 0}{s_{b}}\). Using given values, \(t = \frac{0.023 - 0}{0.004} = 5.75\). So, the test statistic is approximately 5.75.

03

Determine p-value

The p-value is the probability of obtaining the observed data or more extreme data assuming the null hypothesis is true. Given the sample size here is very large, it can safely be assumed that the sampling distribution of the slope is approximately normally distributed. Hence, a Z table or standard statistical software can be used to find the p-value. Generally, a p-value less than 0.05 provides strong evidence against the null hypothesis. In this case, a t-value of 5.75 is very extreme and the corresponding p-value will be virtually zero, providing very strong evidence against the null hypothesis.

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Key Concepts

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

Null Hypothesis

The null hypothesis is a fundamental concept in statistical hypothesis testing, signifying no effect or no relationship between variables. In the context of simple linear regression, the null hypothesis (\text{H}_0) posits that there is no linear relationship between the independent variable (in this case, height) and the dependent variable (log of weekly earnings). Formally, it states that the slope of the regression line, denoted as \(b\), is equal to zero.

If, after conducting the appropriate tests, we find that data does not provide enough evidence to reject the null hypothesis, we would conclude that height has no effect on earnings. It is essential to note that not rejecting \text{H}_0 does not prove it's true but merely indicates insufficient evidence to conclude a relationship exists.

Alternative Hypothesis

The alternative hypothesis contrasts the null hypothesis by suggesting that a particular effect or relationship does exist. For our example of regression involving height and earnings, the alternative hypothesis (\text{H}_1) asserts that there is, in fact, a significant linear relationship between the two. It indicates that the slope of the regression line \(b\) is not equal to zero.

This hypothesis encapsulates our research question - are taller adults likely to earn more? If our tests lead us to reject the null hypothesis, this is the claim we are left to consider. However, we always test the null hypothesis, not the alternative, and any rejection must be based on sufficient statistical evidence.

Test Statistic

The test statistic is a crucial part of the hypothesis testing process used to determine the likelihood that the null hypothesis can be rejected. It's calculated from sample data and used to make a decision about the hypotheses. In simple linear regression, the test statistic for evaluating the slope is typically a t-value computed as \(t = \frac{b - 0}{s_b}\), where \(b\) is the estimated slope from the sample and \(s_b\) is the standard deviation.

In our exercise, the test statistic value of approximately 5.75 was calculated using the provided slope and standard deviation. This value is what we will compare against a critical value or use to find a p-value to assess the validity of the null hypothesis.

P-value

The p-value is a fundamental value which tells us how extreme the observed results are, under the assumption that the null hypothesis is correct. It essentially allows us to determine the significance of our test statistic. If the p-value is less than our chosen significance level (usually 0.05), we have grounds to reject the null hypothesis.

In our regression example, the test statistic was found to be quite high, indicating that the slope is quite different from zero, and therefore, the p-value will be very low. A low p-value, as mentioned, suggests strong evidence against the null hypothesis, implying that the slope is significant, and there likely is a linear relationship between height and the logarithm of weekly earnings.

Linear Relationship

A linear relationship in regression analysis is one where the change in the dependent variable is directly proportional to the change in the independent variable. In simple terms, if we were to plot the relationship between height and log weekly earnings on a graph with a line of best fit if the line slopes upwards or downwards instead of being flat, we're observing a linear relationship.

The strength and direction of this relationship are quantified by the slope (\(b\)) of this line. A nonzero slope suggests a linear relationship exists. In our exercise, we're investigating whether an increase in height is associated with an increase in earnings, which would translate to a positive slope.