Question

The makers of the Lexus EX1274 automobile have steadily increased their sales since their U.S. launch in \(1989 .\) However, the rate of increase changed in 1996 when Lexus introduced a line of trucks. The sales of Lexus from 1996 to 2005 are shown in the table: \({ }^{18}\) $$ \begin{aligned} &\begin{array}{l|rrrrrrrrrrr} \text { Year } & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Sales of thousands } & 80 & 100 & 155 & 180 & 210 & 224 & 234 & 260 & 288 & 303 \end{array}\\\ &\text { vehicles } \end{aligned} $$ a. Plot the data using a scatterplot. How would you describe the relationship between year and sales of Lexus? b. Find the least-squares regression line relating the sales of Lexus to the year being measured? c. Is there sufficient evidence to indicate that sales are linearly related to year? Use \(\alpha=.05\) d. Predict the sales of Lexus for the year 2006 using a \(95 \%\) prediction interval. e. If they are available, examine the diagnostic plots to check the validity of the regression assumptions. f. If you were to predict the sales of Lexus in the year \(2015,\) what problems might arise with your prediction?

Short answer

Answer: Potential issues include extrapolation, unforeseen events or changes in market conditions, and the assumptions of the linear regression model.

Step-by-step solution

1. Scatterplot

Create a scatterplot by plotting the given Years as the x-axis values and the Sales (in thousands) as the y-axis values. Examine the scatterplot to observe the relationship between Years and Sales.

2. Least-Squares Regression Line

Use a statistical software or calculator to find the least-squares regression line that relates the Sales of Lexus to the Year being measured. This will give you a linear equation in the form of \(y = a + b(x)\), where \(y\) represents the sales and \(x\) represents the years.

3. Test for Linear Relationship

Perform a hypothesis test to determine if there's sufficient evidence to indicate a linear relationship between Sales and Year. Here, the null hypothesis is that there's no linear relationship (i.e., a correlation coefficient of 0), while the alternative hypothesis is that a linear relationship exists. Use the significance level of \(\alpha = 0.05\), and calculate the test statistic, \(t\). Use the appropriate degrees of freedom and compare the calculated value to the critical value in a t-distribution table to check for a significant relationship.

4. Predict Sales in 2006

Using the least-squares regression line found in Step 2, predict the sales of Lexus for the year 2006 by replacing \(x\) with the value representing the year 2006. Then, calculate the 95% prediction interval using the given formula to estimate the range of predicted sales for the year 2006.

5. Diagnostic Plots

Examine the diagnostic plots (scatterplot of standardized residuals vs. predicted values, histogram or QQ-plot of the residuals) to check the validity of the regression assumptions. The plots should indicate linearity, constant variance, and normality of the residuals.

6. Issues with Prediction for Year 2015

Discuss the possible issues with predicting sales for the year 2015 based on the given data. Some concerns include extrapolation, unforeseen events or changes in market conditions, and the assumptions of the linear regression model.

Key concepts

Scatterplot

Scatterplots are a valuable visualization tool when it comes to examining the relationship between two quantitative variables. In this context, with years on the x-axis and sales in thousands on the y-axis, a scatterplot allows us to visually analyze how these factors are related. By plotting each pair of data points, we can often see at a glance if there's a pattern or trend. For instance, in this exercise, you might notice that the sales numbers generally increase through the years. This gives an initial visual indication of a positive correlation – as the years go by, sales of Lexus vehicles increase. However, it's crucial not to jump to conclusions solely based on visual cues. Further statistical analysis is necessary to confirm any type of relationship.

Least-Squares Regression

The least-squares regression method is employed to find the best-fit line through the data in your scatterplot. This line aims to minimize the sum of the squared differences (residuals) between the observed data points and the points on the line itself. The equation of the least-squares regression line is usually given as \[ y = a + b(x) \]where

• \(y\) represents the predicted value of the dependent variable (sales, in this case),
• \(x\) is the independent variable (year),
• \(a\) is the y-intercept, and
• \(b\) is the slope of the line.

This model provides a mathematical representation of the trend seen in the scatterplot, allowing us to make predictions about sales for years beyond those in our dataset.

Hypothesis Testing

Hypothesis testing is integral when validating whether a linear relationship exists between two variables. In this particular exercise, the test checks for a linear relationship between the years and Lexus sales figures.The standard steps in hypothesis testing include:

• Setting up the null hypothesis, denoted as \(H_0\), which states there is no linear relationship (i.e., the slope \(b = 0\)).
• The alternative hypothesis, \(H_a\), posits the existence of a linear relationship (i.e., \(b eq 0\)).
• Using a significance level, typically \(\alpha = 0.05\), to decide when to reject \(H_0\).
• Calculating a test statistic, such as the t-statistic, to evaluate \(H_0\).
• Comparing this statistic against critical values in a t-distribution table to determine if the observed relationship is statistically significant.

This process offers a structured approach to determine if our regression line truly reflects a meaningful pattern in the data.

Prediction Interval

Prediction intervals provide a range within which future observations are expected to fall, with a certain level of confidence, typically 95%. When forecasting Lexus sales for 2006, a prediction interval incorporates the least-squares regression line while accounting for the inherent uncertainty in predictions. To compute a prediction interval, you'll usually use a formula that considers both the standard error of the estimate and other elements like sample size. Unlike a confidence interval, which estimates the mean of the dependent variable, a prediction interval provides a range for a single future observation. This is particularly useful when planning strategies based on predicted sales, as it acknowledges the variability and uncertainty inherent in real-world data, thereby offering a more practical view for future planning. It is worth noting that the accuracy of a prediction interval relies on meeting certain regression assumptions, such as linearity and normality of residuals.