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Chapter 8: Problem 40

Exercise 8.26 presents regression output from a model for predicting the heart weight (in g) of cats from their body weight (in kg). The coefficients are estimated using a dataset of 144 domestic cat. The model output is also provided below. $$\begin{array}{rrrrr}\hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\\\\hline \text { (Intercept) } & -0.357 & 0.692 & -0.515 & 0.607 \\ \text { body wt } & 4.034 & 0.250 & 16.119 & 0.000 \\ \hline\end{array}$$ \(s=1.452 \quad R^{2}=64.66 \% \quad R_{a d j}^{2}=64.41 \%\) (a) We see that the point estimate for the slope is positive. What are the hypotheses for evaluating whether body weight is positively associated with heart weight in cats? (b) State the conclusion of the hypothesis test from part (a) in context of the data. (c) Calculate a \(95 \%\) confidence interval for the slope of body weight, and interpret it in context of the data. (d) Do your results from the hypothesis test and the confidence interval agree? Explain.

Short Answer

Expert verified
Body weight is positively associated with heart weight; both hypothesis test and confidence interval support this.

Step by step solution

01

Formulate the Hypotheses

The goal is to evaluate whether body weight is positively associated with heart weight in cats. We set up the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \): \( H_0: \beta_1 = 0 \) (the slope of body weight is zero, indicating no association)\( H_a: \beta_1 > 0 \) (the slope of body weight is positive, indicating a positive association)
02

Evaluate the Hypothesis Test

From the regression output, the p-value for the body weight coefficient is \( \operatorname{Pr}(>|t|) = 0.000 \). This p-value is much smaller than the typical significance level of 0.05, indicating we can reject the null hypothesis \( H_0 \) in favor of the alternative hypothesis \( H_a \). This suggests that there is a positive association between body weight and heart weight.
03

Calculate the 95% Confidence Interval for the Slope

To find the 95% confidence interval for the slope, use the formula:\[ CI = \hat{\beta}_1 \pm z_{\alpha/2} \times SE(\hat{\beta}_1) \]Here, \( \hat{\beta}_1 = 4.034 \), \( SE(\hat{\beta}_1) = 0.250 \), and for a 95% confidence interval with large sample size, \( z_{\alpha/2} = 1.96 \).\[ CI = 4.034 \pm 1.96 \times 0.250 = 4.034 \pm 0.49 \]Thus, the confidence interval is \( (3.544, 4.524) \).
04

Interpret the Confidence Interval

The 95% confidence interval for the slope of body weight \((3.544, 4.524)\) means that we are 95% confident that the true increase in heart weight per kilogram of body weight in cats is between 3.544 g and 4.524 g. Since this interval does not include 0, it further supports the positive association.
05

Compare Hypothesis Test and Confidence Interval Results

The result from the hypothesis test (rejecting \( H_0 \) in favor of \( H_a \)) aligns with the confidence interval, which does not include zero. Both statistical procedures agree, indicating a significant positive association between body weight and heart weight.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about the data we collect. In the context of the given exercise, we wanted to find out if there is a positive association between body weight and heart weight in cats.
To do this, we set up two hypotheses:
  • The null hypothesis (\( H_0 \)) is that there is no association, meaning the slope of body weight is zero.
  • The alternative hypothesis (\( H_a \)) posits that the slope is positive, showing a positive association.
We use the p-value to make our decision. Here, the p-value is very small, specifically 0.000, much less than the significance level of 0.05. This small p-value tells us that the observed data is unlikely under the null hypothesis. Thus, we reject \( H_0 \) and accept \( H_a \), concluding there is indeed a positive association between body and heart weight.
Confidence Interval
A confidence interval gives us a range of values for an estimate, showing how much uncertainty there is in our measurement. For the slope of body weight in this exercise, a 95% confidence interval was calculated.
The formula used is:\[CI = \hat{\beta}_1 \pm z_{\alpha/2} \times SE(\hat{\beta}_1)\] Where:
  • \( \hat{\beta}_1 = 4.034 \) is the slope estimate.
  • \( SE(\hat{\beta}_1) = 0.250 \) is the standard error.
  • \( z_{\alpha/2} = 1.96 \) is the z-score for a 95% confidence level.
This gives us a confidence interval of (3.544, 4.524).
It indicates that we are 95% confident that the true slope lies within this interval. As the interval does not include zero, it supports the positive association observed in the hypothesis testing step.
Association Study
An association study looks for relationships between variables. In the case of linear regression, we examine how an independent variable (like body weight) relates to a dependent variable (such as heart weight). The regression analysis predicts changes in the dependent variable based on changes in the independent variable.
Here, the positive slope we calculated indicates that as body weight increases, heart weight also tends to increase. This relationship helps us understand the biological link between these characteristics in cats. While correlation does not imply causation, a strong association may suggest further investigations into why these traits are linked.
Regression Coefficients
Regression coefficients are key components in a regression analysis, representing the relationship between each independent variable and the dependent variable. In this exercise:
  • The intercept, noted as -0.357, is the predicted heart weight when the body weight is zero. While not directly interpretable in a real-world context since a cat cannot have zero body weight, it serves as a baseline in the model.
  • The coefficient for body weight, 4.034, tells us how much heart weight is expected to change per unit increase in body weight. Thus, for every kilogram increase in body weight, heart weight is expected to rise by about 4.034 grams.
These coefficients are essential as they quantify the relationship and are used to make predictions from the regression model. A positive coefficient for body weight aligns with the earlier hypothesis of a positive association.

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The following regression output is for predicting the heart weight (in g) of cats from their body weight (in \(\mathrm{kg}\) ). The coefficients are estimated using a dataset of 144 domestic cats. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -0.357 & 0.692 & -0.515 & 0.607 \\ \text { body wt } & 4.034 & 0.250 & 16.119 & 0.000 \\ \hline s=1.452 & R^{2}=64.66 \% & \quad R_{a d j}^{2}=64.41 \% \end{array}$$ (a) Write out the linear model. (b) Interpret the intercept. (c) Interpret the slope. (d) Interpret \(R^{2}\). (e) Calculate the correlation coefficient.

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