Question

Refer to the data in Exercise \(12.8,\) relating \(x\), the number of books written by Professor Isaac Asimov, to \(y\), the number of months he took to write his books (in increments of 100). The data are reproduced below. $$ \begin{array}{l|ccccc} \text { Number of Books, } x & 100 & 200 & 300 & 400 & 490 \\ \hline \text { Time in Months, } y & 237 & 350 & 419 & 465 & 507 \end{array} $$ a. Do the data support the hypothesis that \(\beta=0 ?\) Use the \(p\) -value approach, bounding the \(p\) -value using Table 4 of Appendix I or finding the exact \(p\) -value using the \(t\) -Test for the Slope applet. Explain your conclusions in practical terms. b. Use the ANOVA table in Exercise 12.8 , part \(c,\) to calculate the coefficient of determination \(r^{2}\). What percentage reduction in the total variation is achieved by using the linear regression model? c. Plot the data or refer to the plot in Exercise 12.8 , part b. Do the results of parts a and b indicate that the model provides a good fit for the data? Are there any assumptions that may have been violated in fitting the linear model?

Short answer

Answer: To test the hypothesis for the slope, state the null and alternative hypotheses, and use a linear regression analysis to determine the test statistic and p-value. To calculate the coefficient of determination (r^2), first find the correlation coefficient (r), and square it. To analyze the model's fit, consider the results of the hypothesis test, coefficient of determination, and evaluate the assumptions of the linear regression model, such as linearity, independent errors, constant variance, and normality of errors.

Step-by-step solution

(Part a: Test the hypothesis for the slope)

(To perform a hypothesis test for the slope, we need to state our null hypothesis and alternative hypothesis as follows: Null Hypothesis (H0): β = 0 (The number of books has no linear relationship with the number of months) Alternative Hypothesis (H1): β ≠ 0 (The number of books has a linear relationship with the number of months) We'll use a linear regression analysis to determine the test statistic and p-value for this hypothesis test. We can use statistical software, or calculate the test statistic manually, and then compare it with the critical value from the t-distribution table, referred to as Table 4 of Appendix I in the exercise. Since the problem demands the use of the p-value approach, we'll use the p-value test to calculate the exact p-value or find a bound using Table 4 of Appendix I.)

(Part b: Coefficient of Determination)

(To calculate the coefficient of determination, r^2, we first need to find the correlation coefficient, r. This can be found using the formula: r = Σ[(xi - x_mean)(yi - y_mean)] / sqrt( Σ(xi - x_mean)^2 Σ(yi - y_mean)^2 ) Once we have calculated the correlation coefficient, we can find the r^2 value by squaring r: r^2 = r^2 The coefficient of determination (r^2) represents the proportion of variation in y that can be explained by the linear relationship with x. So, we also need to find the percentage reduction in the total variation that is achieved by using the linear regression model.)

(Part c: Analyzing the model's fit)

(To analyze if the model provides a good fit for the data, we can: 1. Look at the results of the hypothesis test from part a: If we reject the null hypothesis, it suggests there is a linear relationship between the number of books and the number of months. 2. Look at the coefficient of determination from part b: Higher r^2 values indicate a better fit of the linear regression model compared to the total variation in the data. 3. Evaluate assumptions: We also need to examine if the assumptions of the linear regression model are met. Key assumptions include linearity, independent errors, constant variance, and normality of errors. Visual examination of the plot can help in identifying whether these assumptions are violated. By considering all these points, we can discuss if the model provides a good fit for the given data.)

Key concepts

Hypothesis Testing

When using linear regression analysis to test relationships between variables, hypothesis testing is a crucial step. It helps determine if the relationship between the independent and dependent variable truly exists or is merely due to random chance. Here, we are examining whether the number of books authored by Professor Isaac Asimov is linearly related to the number of months it takes to write them. We start by formulating the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_1 \)):

• Null Hypothesis (\( H_0 \)): \( \beta = 0 \) (No linear relationship)
• Alternative Hypothesis (\( H_1 \)): \( \beta eq 0 \) (Exists a linear relationship)

To test these hypotheses, a t-test for the slope (\( \beta \)) is conducted. Using statistical software or manual calculation, we determine the p-value, which indicates the probability of observing the data given that the null hypothesis is true. If this p-value is small, typically less than 0.05, we reject \( H_0 \). Rejecting \( H_0 \) suggests that a linear relationship between x (books) and y (months) exists. This aids in making informed decisions based on statistical evidence rather than assumptions.

Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), provides insight into how well a model explains the variability of the outcome data. It is a value ranging from 0 to 1, indicating the proportion of variability in the dependent variable that can be explained by the independent variable. To compute \( r^2 \), we first find the correlation coefficient (\( r \)) using the formula: \[ r = \frac{\Sigma{(x_i - \text{mean}(x))(y_i - \text{mean}(y))}}{\sqrt{\Sigma{(x_i - \text{mean}(x))^2} \cdot \Sigma{(y_i - \text{mean}(y))^2}}}\]Raising \( r \) to the power of 2 gives \( r^2 \): \[ r^2 \]A high \( r^2 \) value shows that a large proportion of variability in the number of months (y) can be explained by the number of books (x). For example, an \( r^2 \) value of 0.85 indicates that 85% of the variation in writing time is attributable to the number of books. This suggests that the linear regression model is an effective tool for understanding the relationship between the variables.

Regression Assumptions

For a linear regression analysis to yield valid results, certain assumptions about the data must be met:

1. Linearity: The relationship between the independent and dependent variables should be linear. This can be evaluated through a scatter plot.
2. Independent Errors: The residuals, or errors, should be independent of each other.
3. Homoscedasticity: The residuals should have constant variance at every level of the independent variable, which can be checked using a plot of residuals against fitted values.
4. Normality of Errors: Residuals should be approximately normally distributed. This can be assessed with a Q-Q plot or a histogram.

By reviewing these assumptions, we ensure the integrity of the regression model. If any assumptions are violated—like if residuals display a pattern—it might mean that a linear model is not the best fit for the data. As shown in the analysis, examining these elements allows us to accurately determine if the linear regression is appropriate for predicting the relationship between the number of books and the time taken to write them.