91Ó°ÊÓ

Bardi Trucking Co., located in Cleveland, Ohio, makes deliveries in the Great Lakes region, the Southeast, and the Northeast. Jim Bardi, the president, is studying the relationship between the distance a shipment must travel and the length of time, in days, it takes the shipment to arrive at its destination. To investigate, Mr. Bardi selected a random sample of 20 shipments made last month. Shipping distance is the independent variable and shipping time is the dependent variable. The results are as follows: $$ \begin{array}{|rcc|ccc|} \hline & \text { Distance } & \text { Shipping Time } & & \text { Distance } & \text { Shipping Time } \\ \text { Shipment } & \text { (miles) } & \text { (days) } & \text { Shipment } & \text { (miles) } & \text { (days) } \\ \hline 1 & 656 & 5 & 11 & 862 & 7 \\ 2 & 853 & 14 & 12 & 679 & 5 \\ 3 & 646 & 6 & 13 & 835 & 13 \\ 4 & 783 & 11 & 14 & 607 & 3 \\ 5 & 610 & 8 & 15 & 665 & 8 \\ 6 & 841 & 10 & 16 & 647 & 7 \\ 7 & 785 & 9 & 17 & 685 & 10 \\ 8 & 639 & 9 & 18 & 720 & 8 \\ 9 & 762 & 10 & 19 & 652 & 6 \\ 10 & 762 & 9 & 20 & 828 & 10 \\ \hline \end{array} $$ a. Draw a scatter diagram. Based on these data, does it appear that there is a relationship between how many miles a shipment has to go and the time it takes to arrive at its destination? b. Determine the correlation coefficient. Can we conclude that there is a positive correlation between distance and time? Use the .05 significance level. c. Determine and interpret the coefficient of determination. d. Determine the standard error of estimate. e. Would you recommend using the regression equation to predict shipping time? Why or why not?

Step by step solution

01

Organize the Data

First, we need to list the shipment distances and their corresponding shipping times from the table provided: \( (656, 5), (853, 14), (646, 6), \dots (828, 10) \). This can be organized comprehensively for plotting and further statistical analysis.

02

Plot the Scatter Diagram

Plot the distances (independent variable) on the x-axis and the shipping time (dependent variable) on the y-axis for each shipment. This visual plot will help us identify any apparent trend or relationship. When plotted, any discernible pattern (like a rising trend) would suggest a relationship between the two variables.

03

Calculate the Correlation Coefficient

The correlation coefficient \( r \) is calculated using the formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \] Where \( n = 20 \), and the sums are calculated from the data points. Substituting the numerical data into the formula provides the result \( r \). If \( r \) is significantly higher than zero, it indicates a positive correlation.

04

Test Significance of the Correlation

Using the calculated \( r \), perform a t-test for significance using:\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \] Compare the calculated \( t \) with the critical value from the t-distribution table at \( \alpha = 0.05 \) . If \( t \) is greater than the critical value, the correlation is significant.

05

Determine the Coefficient of Determination

The coefficient of determination, \( r^2 \), indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. Calculate \( r^2 \) by squaring the correlation coefficient \( r \). \( r^2 \) explains how effectively distance predicts shipping time.

06

Compute the Standard Error of the Estimate

The standard error of the estimate (SEE) is calculated using:\[ SEE = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n-2}} \] Where \( \hat{y}_i \) are the predicted values from the regression line. This measure tells us how accurate the predictions are compared to actual values.

07

Evaluate Use of Regression Equation for Prediction

If \( r \) and \( r^2 \) are strong (close to 1), and the SEE is low, the regression equation is reliable for prediction. If \( r \) is weak (close to 0), predictions may not be reliable. Provide recommendation based on these statistics.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient, often represented as \( r \), is a crucial statistic in regression analysis. It measures the strength and direction of a linear relationship between two variables. In the context of our shipping data, it tells us how closely the distance (miles) and the shipping time (days) are linearly related. When calculating \( r \), we use specific data points to assess how one variable changes with the other.

• If \( r \) is close to 1, it indicates a strong positive correlation, meaning as distance increases, shipping time tends to increase.
• If \( r \) is close to -1, it indicates a strong negative correlation, suggesting that as distance increases, shipping time decreases.
• If \( r \) is near 0, it implies no linear correlation between distance and time.

To determine the significance of \( r \), we conduct a t-test. This involves comparing the calculated t-value with a critical value from the t-distribution table. A significant \( r \) tells us that there is indeed a meaningful relationship between distance and time in our data.

Coefficient of Determination

The coefficient of determination is symbolized as \( r^2 \) and is derived by squaring the correlation coefficient \( r \). It provides a clear picture of the proportion of variance in the dependent variable that can be predicted from the independent variable. For the shipping problem, \( r^2 \) explains how much of the variation in shipping time is attributable to the distance traveled.
A high \( r^2 \) value indicates that a significant portion of the shipping time variability is accounted for by distance. For instance, if \( r^2 = 0.8 \), it suggests that 80% of the variation in shipping time is explained by the distance. Understanding \( r^2 \) helps in assessing the strength and usefulness of the regression model. The nearer \( r^2 \) is to 1, the better the model fits the data. Conversely, a low \( r^2 \) suggests that other factors may be influencing shipping time besides distance.

Standard Error of Estimate

The standard error of estimate (SEE) measures the accuracy of predictions made using the regression equation. It assesses the average distance that the observed values deviate from the regression line.
To compute the SEE, differences between observed values and predicted values (from the regression line) are examined. A lower SEE indicates that the model provides better predictions, as the actual data points lie closer to the regression line.

• A high SEE suggests the model's predictions do not closely follow the actual observed values.
• A low SEE means the predicted shipping times are close to actual shipping times, indicating a reliable model.

Evaluating the SEE assists in determining how well the regression equation fits the data, influencing decisions on using the model for prediction.

Scatter Diagram

A scatter diagram is a graphical representation that showcases the relationship between two variables. For our shipping data, it involves plotting shipment distances on the x-axis and shipping times on the y-axis. Each point on the graph represents a data pair (distance, time).
The scatter diagram helps visually assess whether there is a correlation or trend between distance and time. Patterns can vary:

• A rising pattern suggests a positive correlation, where longer distances often mean longer shipping times.
• A flat distribution implies no correlation, indicating distance does not significantly affect shipping time.

This visual tool is valuable for initial analysis before diving into more complex calculations, like computing \( r \) or \( r^2 \), as it provides immediate insight into potential relationships in the data.