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Chapter 13: Problem 20

Do taller adults make more money? The authors of the paper 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.

Short Answer

Expert verified
Given the slope b = 0.023, standard deviation of b, \(s_{b}\) = 0.004. The calculated t-statistic will be compared to the critical t-value, and this comparison will serve to support or reject the validity of the null hypothesis, depending on the outcome.

Step by step solution

01

Understanding the Statements and Formulating Hypotheses

First, review the given parameters. In this case, the slope of the estimated regression line, b, is 0.023 and the standard deviation of b, \(s_{b}\), is 0.004. The hypotheses can be written as: \n\nNull Hypothesis: \(H_{0}: b = 0\) \n\nAlternative Hypothesis: \(H_{a}: b \neq 0\)
02

Calculation of t-statistic

Next, calculate the t-statistic, which is essentially the ratio of the estimated parameter to its standard deviation. \n\nUsing the formula for t-statistic: \[t = \frac{b}{s_{b}}\]\n\nSubstitute the given values into the formula: \[t = \frac{0.023}{0.004}\]
03

Compare t-statistic to critical value

The calculated t-statistic must now be compared to the t-critical value from the t-distribution table. The usual significance level is 0.05 (5%). Assume there are large degrees of freedom due to the large sample size, the critical value for a two-tailed test is approximately 1.96. If the calculated t-statistic is greater than the t-critical value, it provides convincing evidence to reject the null hypothesis.
04

Conclusion of the Test

Based on the comparison of the calculated t-statistic and the critical t-value, you make a final decision about the hypothesis. If the t-statistic is less than the t-critical value, fail to reject the null hypothesis. If the t-statistic is greater than the t-critical value, reject the null hypothesis, which supports the claim of a useful linear relationship between height and earnings.

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Key Concepts

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

Simple Linear Regression
Simple linear regression is a statistical method used to understand the relationship between two continuous variables. In the context of this exercise, it explores how height (measured in inches) affects earnings (expressed as the logarithm of weekly gross earnings in dollars). Here, height is the independent variable, while earnings are the dependent variable. The goal is to determine how a change in height influences a change in earnings.

The regression line is described by the equation: \[y = a + bx\]where:
  • \(y\) is the dependent variable (log earnings),
  • \(x\) is the independent variable (height),
  • \(a\) is the y-intercept, and
  • \(b\) is the slope of the line, representing the change in \(y\) for each unit change in \(x\).
The slope \(b = 0.023\) indicates that for each additional inch in height, the log of weekly earnings increases by 0.023, assuming all other factors remain constant. This technique is powerful for predicting and explaining relationships between variables, providing insights into data trends.
T-statistic
The t-statistic is a key element in hypothesis testing within regression analysis. It measures how many standard deviations the estimated parameter (the slope \(b\), in this case) is away from the value stated in the null hypothesis.

To calculate the t-statistic, use the formula:\[t = \frac{b}{s_b}\]where:
  • \(b\) is the estimated slope (0.023), and
  • \(s_b\) is the standard deviation of \(b\) (0.004).
By substituting the given values into the formula, we find:
\[t = \frac{0.023}{0.004} = 5.75\]

The t-statistic (5.75) indicates how much more extreme the observed slope is, compared to what we would expect if the true slope were zero. A high absolute value suggests a significant effect, providing evidence against the null hypothesis. The process involves comparing the t-statistic with a critical value from statistical tables to determine if the relationship is genuinely significant.
Null Hypothesis
The null hypothesis is a fundamental concept in hypothesis testing. It represents the default or initial claim made about the data, which is often the perspective of no effect or no relationship.

In our scenario of examining the relationship between height and earnings, the null hypothesis is:\[H_0: b = 0\]This statement suggests that there is no linear relationship between height and earnings (i.e., the slope \(b\) is zero).

Hypothesis testing involves checking this initial hypothesis against the data, to see if there is enough evidence to reject it. Here, the aim is to determine if the observed slope of the regression line is significantly different from zero. A rejection of the null hypothesis indicates a statistically significant relationship. Otherwise, we conclude that any observed effect could reasonably occur by chance, and height might not reliably predict earnings. This testing is crucial for validating the assumptions made by researchers and supporting their findings with statistical evidence.

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Most popular questions from this chapter

A simple linear regression model was used to describe the relationship between sales revenue \(y\) (in thousands of dollars) and advertising expenditure \(x\) (also in thousands of dollars) for fast-food outlets during a 3 -month period. A sample of 15 outlets yielded the accompanying summary quantities. $$ \begin{aligned} &\sum x=14.10 \quad \sum y=1438.50 \quad \sum x^{2}=13.92 \\ &\sum y^{2}=140,354 \quad \sum x y=1387.20 \\ &\sum(y-\vec{y})^{2}=2401.85 \quad \sum(y-\hat{y})^{2}=561.46 \end{aligned} $$ a. What proportion of observed variation in sales revenue can be attributed to the linear relationship between revenue and advertising expenditure? b. Calculate \(s_{e}\) and \(s_{b}\) c. Obtain a \(90 \%\) confidence interval for \(\beta\), the average change in revenue associated with a \(\$ 1000\) (that is, 1 -unit) increase in advertising expenditure.

The paper suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length \((\mathrm{mm})\) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation \(0.005\). a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m}\) ? c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) has a shell thickness of greater than .175? Less than . 178 ?

If the sample correlation coefficient is equal to 1, is it necessarily true that \(\rho=1\) ? If \(\rho=1\), is it necessarily true that \(r=1 ?\)

An investigation of the relationship between \(x=\) traffic flow (thousands of cars per 24 hours) and \(y=\) lead content of bark on trees near the highway (mg/g dry weight) yielded the accompanying data. A simple linear regression model was fit, and the resulting estimated regression line was \(\hat{y}=28.7+33.3 x .\) Both residuals and standardized residuals are also given. \(\begin{array}{lrrrrr}x & 8.3 & 8.3 & 12.1 & 12.1 & 17.0 \\ y & 227 & 312 & 362 & 521 & 640 \\ \text { Residual } & -78.1 & 6.9 & -69.6 & 89.4 & 45.3 \\ \text { St. resid. } & -0.99 & 0.09 & -0.81 & 1.04 & 0.51 \\ x & 17.0 & 17.0 & 24.3 & 24.3 & 24.3 \\ y & 539 & 728 & 945 & 738 & 759 \\ \text { Residual } & -55.7 & 133.3 & 107.2 & -99.8 & -78.8 \\ \text { St. resid. } & -0.63 & 1.51 & 1.35 & -1.25 & -0.99\end{array}\) a. Plot the \((x\), residual \()\) pairs. Does the resulting plot suggest that a simple linear regression model is an appropriate choice? Explain your reasoning. b. Construct a standardized residual plot. Does the plot differ significantly in general appearance from the plot in Part (a)?

Sea bream are one type of fish that are often raised in large fish farming enterprises. These fish are usually fed a diet consisting primarily of fish meal. The authors of the paper describe a study to investigate whether it would be more profitable to substitute plant protein in the form of sunflower meal for some of the fish meal in the sea bream's diet. The accompanying data are consistent with summary quantities given in the paper for \(x=\) percentage of sunflower meal in the diet and \(y=\) average weight of fish after 248 days (in grams). \begin{tabular}{cc} Sunflower Meal (\%) & Average Fish Weight \\ \hline 0 & 432 \\ 6 & 450 \\ 12 & 455 \\ 18 & 445 \\ 24 & 427 \\ 30 & 422 \\ 36 & 421 \\ \hline \end{tabular} The estimated regression line for these data is \(\hat{y}=448.536-0.696 x\) and the standardized residuals are as given. \begin{tabular}{cc} Sunflower Meal (\%), \(x\) & Standardized Residual \\ \hline 0 & \(-1.96\) \\ 6 & \(0.58\) \\ 12 & \(1.42\) \\ 18 & \(0.84\) \\ 24 & \(-0.46\) \\ 30 & \(-0.58\) \\ 36 & \(-0.29\) \\ \hline \end{tabular} Construct a standardized residual plot. What does the plot suggest about the adequacy of the simple linear regression model?
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