Question

Refer to the data in Exercise 11 (Section 12.2), 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. Explain your conclusions in practical terms. b. Construct the ANOVA table or use the one constructed in Exercise 11 (Section 12.2), 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 11 (Section 12.2), 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

In conclusion, the results from parts a and b show that there is a significant relationship between the number of books written by Professor Asimov and the time taken to write them, with the model explaining 68% of the variation in the time spent writing books. However, due to the small sample size, data increments of 100, and possible violation of some assumptions (normality and constant variance), we should be cautious in interpreting the results and consider additional factors or a more complex model that could better explain the relationship between the number of books and the time spent writing them.

Step-by-step solution

Calculate the test statistic t

Using the formula: $$ t = \frac{\hat{\beta} - 0}{\text{SE}(\hat{\beta})} $$ We first need to compute the estimate of the standard error, \(\text{SE}(\hat{\beta})\), and the estimated slope, \(\hat{\beta}\). Using a statistical software or calculator, enter the given data to find: $$ \hat{\beta} = 1.553, \qquad \text{SE}(\hat{\beta}) = 0.297 $$ Now, we can calculate the value of our test statistic \(t\): $$ t = \frac{1.553 - 0}{0.297} = 5.23 $$

Calculate the approximate p-value

Now that we have the test statistic, \(t = 5.23\), we need to find the degrees of freedom: $$ \text{df} = n - 2 = 5 - 2 = 3 $$ With the given degrees of freedom (3), we can use Table 4 of Appendix I to find the approximate p-value. Since the test statistic is positive (\(t = 5.23\)), we need to look in the right tail. We can determine that \(0.01 < p < 0.025\).

Interpret the results

Since the \(p\)-value is very small (\(0.01 < p < 0.025\)), we reject the null hypothesis. This means that we have strong evidence that the number of books written by Professor Asimov and the time taken to write them are related. The slope of the regression line is not equal to 0, indicating that there is a significant relationship between the number of books and the time spent writing them. b) To calculate the coefficient of determination \(r^{2}\), we can use the output obtained from the statistical software or calculator.

Calculate the regression sum of squares (SSR) and the total sum of squares (SST)

Using the output from the statistical software or calculator, we find: $$ \text{SSR} = 14640.8, \qquad \text{SST} = 21434.0 $$

Calculate the coefficient of determination r^2

The coefficient of determination is calculated as follows: $$ r^{2} = \frac{\text{SSR}}{\text{SST}} = \frac{14640.8}{21434.0} = 0.6828 $$ Rounding this to 2 decimal places, we get \(r^{2} = 0.68\). This means that \(68 \%\) of the variation in the time spent writing books can be explained by the number of books written. Consequently, using the linear regression model leads to a \(68 \%\) reduction in the total variation. c) Referring back to the plot from Exercise 11 in Section 12.2, we can see that there seems to be a positive linear relationship between the number of books written and the time taken to write them. From parts a) and b), we found evidence that this relationship is significant and that the model explains \(68 \%\) of the variation in the time spent writing books. However, since the sample size is small and there may be other factors influencing the writing time, the linear model may not capture all the complexity of the relationship. Additionally, since the data is given in increments of \(100\), there might be some inherent violation of the normality and constant variance assumptions used in the linear regression model.

Key concepts

Hypothesis Testing

In the realm of statistics, hypothesis testing serves as a fundamental method for making inferences about population parameters using sample data. Particularly in linear regression, it examines whether the independent variable has a statistically significant effect on the dependent variable. For instance, in the exercise, the hypothesis being tested is whether the slope coefficient \( \beta \) is zero, implying no relationship between the numbers of books Professor Asimov wrote (independent variable) and the time he took to write them (dependent variable).

The test involves calculating a test statistic, here represented as a t-score, which quantifies the degree to which the sample estimate deviates from the null hypothesis value. This t-score is then used to find a corresponding p-value which indicates the probability of observing such an extreme test statistic under the null hypothesis. Generally, a low p-value (commonly below 0.05) leads to the rejection of the null hypothesis, implying that the relationship in the population is likely to be non-zero.

ANOVA

ANOVA, which stands for Analysis of Variance, is another statistical technique often used in the context of regression to assess the overall fit of a model. It partitions the total variation in the dependent variable into variation explained by the model (regression sum of squares) and unexplained random variation (error sum of squares).

In simpler terms, when we're looking at linear regression, ANOVA compares the variance captured by the regression line with the total variance observed in the data. Through an ANOVA table, researchers can understand how well the independent variables, collectively, explain the variation in the dependent variable. The F-statistic, derived from the ANOVA table, further helps in testing the overall significance of the regression model.

Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), plays a pivotal role in depicting the strength and quality of a regression model. Simply put, it's a statistical measure that represents the proportion of the variance in the dependent variable that's predictable from the independent variable(s).

For any model, \( r^2 \) values range from 0 to 1, where 0 signifies no explanatory power and 1 indicates perfect prediction. A higher \( r^2 \) value means greater variance reduction and typically infers a model that fits the data better. Reflecting on the provided exercise, the \( r^2 \) value of 0.68 tells us that 68% of the variation in writing time is explainable by the number of books written—a substantial amount, indicative of a strong linear relationship.

P-value Approach

The p-value approach is a technique widely used to draw conclusions in hypothesis testing. This approach involves determining the smallest level of significance at which the null hypothesis would be rejected, based on the observed data.

A small p-value typically means that the observed data would be unlikely if the null hypothesis were true, leading to the rejection of the null hypothesis. For instance, in the exercise, the calculated p-value falls between 0.01 and 0.025, which is substantially lower than the commonly used significance level of 0.05. This leads to the conclusion that the relationship between the number of books written and the time spent is indeed statistically significant.

Regression Analysis

Regression analysis is a powerful statistical tool for investigating the relationship between variables. It not only estimates the coefficients that characterise the relationship between the dependent and independent variables but also enables forecasting, prediction, and hypothesis testing.

In linear regression, we aim to fit a line through the observed data points that minimises the sum of the squared differences between the observed values and those predicted by the model. It's important to note that an effective regression analysis often depends on meeting certain assumptions—like the linearity, independence, and normal distribution of the residuals—which, if violated, may lead to incorrect conclusions.