91Ó°ÊÓ

A regression analysis carried out to relate \(y=\) repair time for a water filtration system ( \(\mathrm{hr}\) ) to \(x_{1}=\) elapsed time since the previous service (months) and \(x_{2}=\) type of repair ( 1 if electrical and 0 if mechanical) yielded the following model based on \(n=12\) observations: \(y\) \(=.950+.400 x_{1}+1.250 x_{2}\). In addition, SST \(=12.72, \mathrm{SSE}=2.09\), and \(s_{\hat{\beta}_{2}}=.312\). a. Does there appear to be a useful linear relationship between repair time and the two model predictors? Carry out a test of the appropriate hypotheses using a significance level of \(.05\). b. Given that elapsed time since the last service remains in the model, does type of repair provide useful information about repair time? State and test the appropriate hypotheses using a significance level of \(.01\). c. Calculate and interpret a 95\% CI for \(\beta_{2}\). d. The estimated standard deviation of a prediction for repair time when elapsed time is 6 months and the repair is electrical is .192. Predict repair time under these circumstances by calculating a \(99 \%\) prediction interval. Does the interval suggest that the estimated model will give an accurate prediction? Why or why not?

Step by step solution

01

Establish Hypotheses for Global Test (Part a)

First, determine the null and alternative hypotheses for checking the overall model significance. The null hypothesis (H0) is that there is no linear relationship, i.e., all the coefficients of the predictors are zero: \[ H_0: \beta_1 = \beta_2 = 0 \]The alternative hypothesis (H1) is that at least one of the coefficients is not zero:\[ H_1: \text{At least one } \beta_i eq 0 \]

02

Calculate F-statistic for Global Test (Part a)

Use the formula for the F-statistic:\[ F = \frac{(SST - SSE) / p}{SSE / (n - p - 1)} \]where SST = 12.72, SSE = 2.09, n = 12, and p = 2 (number of predictors). Compute:\[ F = \frac{(12.72 - 2.09)/2}{2.09/9} = \frac{10.63/2}{0.2322} = 22.87 \]

03

Decision for Global Test (Part a)

Compare the computed F-statistic to the critical value from the F-distribution table at \( \alpha = 0.05\), with 2 and 9 degrees of freedom. Given that F-critical is around 4.26, and since 22.87 > 4.26, we reject the null hypothesis. This indicates a significant linear relationship between the predictors and repair time.

04

Establish Hypotheses for Type of Repair (Part b)

For testing whether type of repair provides useful information (\(x_2\)), set the hypotheses. Null hypothesis \(H_0\) assumes \(\beta_2 = 0\), indicating type of repair provides no useful information. The alternative hypothesis \(H_1\) assumes \(\beta_2 eq 0\).

05

Calculate t-statistic for Type of Repair Test (Part b)

Use the t-statistic formula:\[ t = \frac{\hat{\beta_2}}{s_{\hat{\beta_2}}} = \frac{1.250}{0.312} = 4.006 \]

06

Decision for Type of Repair Test (Part b)

Determine the critical t-value for a two-tailed test with \(\alpha=0.01\) and 9 degrees of freedom (since t-critical ≈ ±3.25). Since 4.006 > 3.25, we reject \(H_0\). The type of repair is a significant predictor.

07

Calculate Confidence Interval (CI) for \(\beta_2\) (Part c)

Calculate the 95% CI for \(\beta_2\):\[ 1.250 \pm t^* \times 0.312 \]Using t-value ≈ 2.262 (from t-distribution table with 9 df), the CI is:\[ 1.250 \pm 2.262 \times 0.312 = 1.250 \pm 0.705 \]Thus, CI is \([0.545, 1.955]\). This suggests \(\beta_2\) could realistically lie within this range.

08

Calculate Prediction Interval (PI) for Repair Time (Part d)

First calculate the predicted value when \(x_1 = 6\) and \(x_2 = 1\):\[ y = 0.950 + 0.400(6) + 1.250(1) = 0.950 + 2.400 + 1.250 = 4.6 \]Now, calculate the 99% PI using the formula:\[ y \pm t^* \cdot \sigma_{pred} = 4.6 \pm 3.250 \cdot 0.192 \]where t-value ≈ 3.250:\[ 4.6 \pm 0.624 = [3.976, 5.224] \]The interval is relatively narrow, suggesting a reasonably accurate prediction.

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Key Concepts

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

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between a dependent variable and one or more independent variables. The goal is to establish the best-fitting line (or hyperplane in multiple dimensions) that describes how the dependent variable changes as the independent variables change. In our case, we're modeling the repair time for a water filtration system based on the elapsed time since the previous service and the type of repair.
In linear regression, the equation is often written as \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon \]

• \(y\) is the dependent variable (repair time).
• \(x_1\) and \(x_2\) are the independent variables (elapsed time and type of repair).
• \(\beta_0, \beta_1, \beta_2\) are coefficients determined by the regression analysis.
• \(\epsilon\) represents the error term or unexplained variation.

The coefficients indicate the expected change in the dependent variable for a one-unit change in an independent variable, assuming all other variables remain constant. It provides a way to predict the repair time based on prior service time and type of repair.

Hypothesis Testing

Hypothesis testing in regression analysis involves making inferences about the relationship between variables by testing whether the coefficients of the independent variables are significantly different from zero. This tests the null hypothesis, \(H_0\), that states there is no relationship between the dependent and independent variables.
In our analysis:

• For the overall model: \(H_0: \beta_1 = \beta_2 = 0\).
• Alternative hypothesis \(H_1\), that at least one \(\beta\) is not zero, suggesting a significant linear relationship.

By calculating the F-statistic, we assess the overall fit of the model. If the computed F-statistic is greater than the critical value from the F-distribution table, we reject the null hypothesis, indicating the predictors have a significant relationship with the dependent variable. For individual coefficients, a t-statistic is used to determine the significance of each predictor. If the t-statistic for a coefficient exceeds the critical t-value, the null hypothesis for that predictor is rejected.

Confidence Interval

Confidence intervals provide a range of values for the estimated coefficient that is believed to contain the true population parameter with a certain level of confidence, often 95%.
For example, calculating a 95% confidence interval for the coefficient \(\beta_2\) (type of repair) gives us an idea of the range where the true value of \(\beta_2\) might lie. We calculate it using the formula: \[ \hat{\beta_2} \pm t^* \times s_{\hat{\beta_2}} \] where \(t^*\) is the critical t-value.
This interval provides insight into the reliability and precision of the coefficient estimate.

• If the interval includes zero, it suggests the predictor might not be significant.
• If it does not include zero, this suggests the predictor has a real effect on the dependent variable.

Confidence intervals help quantify the uncertainty associated with sample estimates, aiding better decision-making.

Prediction Interval

A prediction interval provides a range in which we expect a single new observation to fall, with a given level of confidence. In contrast to confidence intervals that estimate the range for the mean value of the dependent variable, prediction intervals account for both the error in estimating the mean and the variability around that mean for individual observations.
Prediction intervals are generally wider than confidence intervals because they incorporate more sources of uncertainty. For example, to predict the repair time when elapsed time is 6 months and the repair is electrical, the prediction interval might use:\[ y \pm t^* \cdot \sigma_{pred} \]This interval allows us to gauge how well the model might predict an individual future observation, providing a realistic range for expectations.

F-statistic

The F-statistic is a crucial element in regression analysis used to assess whether the overall regression model is a good fit for the data. It is derived from an F-test, which compares the model with no predictors against the model with predictors to determine if the added complexity is statistically warranted.
The calculation involves comparing the model's systematic variance with its unsystematic variance:\[ F = \frac{(SST - SSE) / p}{SSE / (n - p - 1)}\] where:

• \(SST\) is the total sum of squares.
• \(SSE\) is the error sum of squares.
• \(n\) is the number of observations.
• \(p\) is the number of predictors.

A high F-statistic relative to the critical value suggests that the predictors explain a significant portion of the variance in the dependent variable.

T-statistic

The t-statistic in regression analysis helps determine whether a specific predictor is significantly contributing to the model. It does so by testing if the regression coefficient for a predictor is significantly different from zero.
The formula for calculating the t-statistic is:\[ t = \frac{\hat{\beta}}{s_{\hat{\beta}}} \] where \(\hat{\beta}\) is the estimated coefficient and \(s_{\hat{\beta}}\) is its standard error.

• Compare the calculated t-statistic to a critical t-value from the t-distribution table, usually based on a 95% confidence level.
• If the t-statistic is larger than the critical value, reject the null hypothesis for that predictor.

The t-statistic is vital for assessing the significance of individual predictors, determining which contribute meaningfully to the explanatory power of the model.