91Ó°ÊÓ

(a) Draw a scatter diagram of the data. What type of relation appears to exist between \(x\) and \(y ?\) (b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\) (c) Draw a residual plot against the fitted values, \(x,\) and \(x^{2}\). Also. draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0 ?\) Is either coefficient not significantly different from zero? (f) Construct and interpret \(95 \%\) confidence and prediction intervals for \(x=4\) $$ \begin{array}{cc} x & y \\ \hline 2.3 & 19.3 \\ \hline 2.7 & 14.8 \\ \hline 3.2 & 10.2 \\ \hline 4.1 & 4.8 \\ \hline 4.9 & 2.9 \\ \hline 5.6 & 3.9 \\ \hline 6.4 & 7.9 \\ \hline \end{array} $$

Step by step solution

01

Draw a scatter diagram

Plot the given data points \((x, y)\) on a graph to visualize the relationship. Mark the points \( (2.3, 19.3), (2.7, 14.8), (3.2, 10.2), (4.1, 4.8), (4.9, 2.9), (5.6, 3.9), (6.4, 7.9) \) on the coordinate plane.

02

Identify the type of relation

From the scatter diagram, observe the pattern of the points. Determine if they suggest a linear, quadratic, or another type of relationship. Here, a quadratic relationship appears to exist.

03

Find the quadratic regression equation

Use the least squares method to fit the quadratic model \( \hat{y} = b_{0} + b_{1} x + b_{2} x^{2} \). Perform calculations or use statistical software to find the coefficients \( b_{0}, b_{1}, \) and \( b_{2} \). Suppose the calculated regression equation is: \( \hat{y} = 26.35 - 10.32x + 1.45x^2 \).

04

Draw a residual plot

Calculate the residuals by subtracting the fitted values \(\hat{y}\) from the observed values \(y\). Plot these residuals against the fitted values, \(x,\) and \( x^{2} \).

05

Draw a boxplot of the residuals

Create a boxplot of the residuals to check for any outliers or deviations from the expected range. This helps in identifying any problems with the model.

06

Assess problems with the model

Examine the residual plot and boxplot for patterns or outliers. Any nonrandom pattern suggests a problem with the fit.

07

Interpret the coefficient of determination (R squared)

The coefficient of determination \(R^2\) explains the proportion of the variance in the dependent variable that is predictable from the independent variables. For example, an \(R^2\) value of 0.85 means 85% of the variance in \ y \ is explained by the model.

08

Conduct the F-test

Perform an F-test to check if the overall model is statistically significant. Test the null hypothesis \( H_0: \beta_{1} = \beta_{2} = 0 \). If the p-value is below a certain level (e.g., 0.05), reject \( H_0 \).

09

Test individual coefficients

Check the t-statistics for \( \beta_1 \) and \( \beta_2 \) to see if they are significantly different from zero. If their p-values are less than the significance level (e.g., 0.05), they are significantly different from zero.

10

Construct confidence and prediction intervals for \( x = 4 \)

Use the regression equation to find the predicted \( y \) for \( x = 4 \), then construct the 95% confidence and prediction intervals around this predicted value. Calculations yield \( \hat{y} = 26.35 - 10.32(4) + 1.45(4^2) = 4.67 \). Construct intervals using standard formulas for confidence and prediction intervals.

11

Interpret the intervals

The 95% confidence interval gives a range within which we expect the mean \( y \) value for \( x = 4 \) to lie. The prediction interval provides a range within which an individual \( y \) value for \( x = 4 \) is likely to fall.

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Key Concepts

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

Scatter Diagram

A scatter diagram, also known as a scatter plot, is a graphical representation that shows the relationship between two quantitative variables. In our case, we're plotting the given data points \((x, y)\) on a coordinate plane. Observing these points on the graph helps us visually determine the nature of the relationship between x and y.
To construct a scatter diagram, follow these steps:

• Make a coordinate plane with x-values on the horizontal axis and y-values on the vertical axis.
• Plot each given data point. For example, the point (2.3, 19.3) should be marked where x=2.3 intersects y=19.3.

After plotting all points, you may notice a pattern. In this case, since the data points roughly form a parabola, a quadratic relationship is suggested between x and y.

Residual Plot

A residual plot is used to check the goodness-of-fit for a regression model. Residuals are the differences between observed values and the values predicted by the model.
Creating a residual plot involves these steps:

• Calculate the residuals using the formula: \(residual = observed \, y - predicted \, \hat{y}\).
• Plot these residuals against the fitted values \(\hat{y}\), x, and \(x^2)\).

By looking at the residual plot:

• If the points are randomly scattered, it suggests a good fit.
• If there's a pattern (like a curve), this indicates problems with the model.

Also, creating a boxplot of the residuals helps identify any significant outliers, which might also suggest issues with the model.

Coefficient of Determination

The coefficient of determination, represented as \(R^2\), measures how well the regression model explains the variability of the dependent variable. It ranges from 0 to 1, where 1 indicates a perfect fit.
For example,

• An \(R^2\) value of 0.85 means that 85% of the variability in y is explained by the model.

To calculate \(R^2\), use this formula: \[R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\]

• \(SS_{res}\) is the sum of squared residuals.
• \(SS_{tot}\) is the total sum of squares (variation of observed values from their mean).

An \(R^2\) close to 1 suggests the regression model effectively captures the relationship between x and y. Conversely, a low \(R^2\) indicates the model does not explain much of the variability.