91Ó°ÊÓ

Does exposure to air pollution result in decreased life expectancy? This question was examined in the article "Does Air Pollution Shorten Lives?" (Sratistics and Public Policy, Reading, MA, Addison-Wesley, 1977). Data on $$ \begin{aligned} &y=\text { total mortality rate (deaths per } 10,000) \\ &x_{1}=\text { mean suspended particle reading }\left(\mu \mathrm{g} / \mathrm{m}^{5}\right) \\ &x_{2}=\text { smallest sulfate reading }\left(\left[\mu \mathrm{g} / \mathrm{m}^{3}\right] \times 10\right) \\ &x_{3}=\text { population density }\left(\text { people } / \mathrm{mi}^{2}\right) \\ &x_{4}=(\text { percent nonwhite }) \times 10 \\ &x_{5}=(\text { percent over } 65) \times 10 \end{aligned} $$ for the year 1960 was recorded for \(n=117\) randomly selected standard metropolitan statistical areas. The estimated regression equation was $$ \begin{aligned} y=& 19.607+.041 x_{1}+.071 x_{2} \\ &+.001 x_{3}+.041 x_{4}+.687 x_{5} \end{aligned} $$ a. For this model, \(R^{2}=.827\). Using a .05 significance level, perform a model utility test. b. The estimated standard deviation of \(\hat{\beta}_{1}\) was 016 . Calculate and interpret a \(90 \%\) CI for \(\beta_{1}\). c. Given that the estimated standard deviation of \(\hat{\beta}_{4}\) is .007, determine whether percent nonwhite is an important variable in the model. Use a 01 significance level. d. In 1960 , the values of \(x_{1}, x_{2}, x_{3}, x_{4}\), and \(x_{5}\) for Pittsburgh were \(166,60,788,68\), and 95 , respectively. Use the given regression equation to predict Pittsburgh's mortality rate. How does your prediction compare with the actual 1960 value of 103 deaths per 10,000 ?

Step by step solution

01

Understanding the Model Utility Test

To perform the model utility test, we assess whether the model explains a significant portion of the variability in the response variable \(y\). Given \( R^2 = 0.827 \), it suggests that 82.7% of the variability in deaths is explained by the model. We will use an \( F \)-test for this purpose.

02

Calculate F-statistic for Model Utility

The \( F \)-statistic is calculated as \( F = \frac{(R^2 / k)}{((1-R^2)/(n-k-1))} \), where \(k\) is the number of predictors and \(n\) is the sample size. Plugging in the given values, \( F = \frac{(0.827 / 5)}{((1 - 0.827) / (117 - 5 - 1))} \). Compute this to get the \( F \)-value.

03

Interpret F-statistic

Compare the calculated \( F \)-statistic with the critical \( F \)-value from \( F \)-distribution tables at \( \alpha = 0.05 \), with \( 5 \) degrees of freedom for the numerator and \( 111 \) degrees of freedom for the denominator. If the calculated \( F \) is greater than the critical \( F \)-value, the model is significant.

04

Confidence Interval for \( \beta_1 \)

The 90% confidence interval for \( \beta_1 \) is given by \( \hat{\beta}_1 \pm t_{\alpha/2} \, \times \, \text{SE}(\hat{\beta}_1) \). With \( \hat{\beta}_1 = 0.041 \) and \( \text{SE}(\hat{\beta}_1) = 0.016 \), use \( t_{0.05,111} \) (value from t-table) to compute the interval.

05

Significance of \( \beta_4 \)

To evaluate the significance of percent nonwhite, compute the t-statistic: \( t = \hat{\beta}_4 / \text{SE}(\hat{\beta}_4) = 0.041 / 0.007 \). Compare this t-value against the critical t-value from the t-distribution table with \( 1\)% significance and \( 111 \) degrees of freedom.

06

Predicting Pittsburgh's Mortality Rate

Substitute the values of \( x_1, x_2, x_3, x_4, x_5 \) for Pittsburgh into the regression equation: \( y = 19.607 + 0.041(166) + 0.071(60) + 0.001(788) + 0.041(68) + 0.687(95) \). Calculate \( y \) to get the predicted mortality rate.

07

Compare Prediction with Actual Value

Once you have the predicted mortality rate for Pittsburgh, compare it with the actual rate of 103 deaths per 10,000. Discuss the closeness and possible reasons for any discrepancies, considering model limitations or missing variables.

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Key Concepts

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

Model Utility Test

The model utility test helps us determine how well a multiple linear regression model explains the variability in the response variable. In our exercise on air pollution and mortality rates, we use this test to evaluate the significance of our model. This is done using the coefficient of determination, denoted as \(R^2\). An \(R^2\) value of 0.827 means 82.7% of the variation in total mortality rate is explained by the predictors in the model. This is a strong indicator that the model is likely useful.

To confirm the model's utility, we conduct an \(F\)-test. We calculate the \(F\)-statistic as:\[F = \frac{(R^2 / k)}{((1-R^2)/(n-k-1))}\]where \(k\) is the number of predictors (5 in this case) and \(n\) is the sample size (117).

• Insert these values into the formula
• Compute the \(F\)-statistic
• Compare it to the critical \(F\)-value from \(F\)-distribution tables

If the calculated \(F\)-value surpasses the critical \(F\)-value at a 0.05 significance level, we conclude that the model provides a significant explanation of the variability in the mortality rate data.

Confidence Interval

A confidence interval provides a range of values within which we expect the true value of a regression coefficient to fall. In the exercise, we're interested in the 90% confidence interval for \(\beta_1\), the coefficient of mean suspended particle reading. This interval helps us understand the precision of our estimate for \(\beta_1\) and how sensitive our model might be to changes in particle readings.

To calculate this, use the formula:\[\hat{\beta}_1 \pm t_{\alpha/2} \times \text{SE}(\hat{\beta}_1)\]where \(\hat{\beta}_1 = 0.041\) and \(\text{SE}(\hat{\beta}_1) = 0.016\). We look up the critical \(t\)-value \(t_{0.05,111}\) for a 90% confidence level in \(t\)-distribution tables.

Calculate the interval by:

• Multiplying the critical \(t\) by the standard error
• Adding/subtracting this product from \(\hat{\beta}_1\)

This range gives us reasonable certainty about where the true \(\beta_1\) lies, enhancing our understanding of its influence on mortality rates.

Significance Testing

Significance testing helps us identify the impact of individual predictors in a regression model. In our study, we examine the significance of \(\beta_4\), the coefficient for percent nonwhite, using a 1% significance level. This step tells us if this predictor has a meaningful contribution to explaining the variation in mortality rates.

To perform this test, compute the \(t\)-statistic as:\[t = \frac{\hat{\beta}_4}{\text{SE}(\hat{\beta}_4)} = \frac{0.041}{0.007}\]Compare this \(t\)-value against the critical \(t\)-value from the \(t\)-distribution table, which gives you the threshold \(t\) for 111 degrees of freedom at the 1% significance level.

• If \(t\) is larger than the critical value, the variable is significant.
• If smaller, it's not significant in explaining the variability.

This test informs us whether considering percent nonwhite provides valuable insights in modeling mortality rates.

Regression Equation

The regression equation is the backbone of predictive analysis within multiple linear regression. It allows us to estimate the response variable based on several predictors. In this exercise, the regression equation is used to predict total mortality rates for 1960 in various metropolitan areas based on air pollution and demographic data.
The given regression equation is:\[y = 19.607 + 0.041x_1 + 0.071x_2 + 0.001x_3 + 0.041x_4 + 0.687x_5\]where each \(x\) corresponds to predictors like mean suspended particle reading or percent nonwhite. This equation assigns a coefficient to each predictor, showing the expected change in mortality rate for a one-unit change in the predictor, all else being constant.
For predictive tasks:

• Substitute known predictors into the equation to calculate the expected response (mortality rate)
• Use this regression model to assess different scenarios or forecast outcomes for new data

In the case of Pittsburgh in 1960, by inserting local values for \(x_1, x_2, x_3, x_4, x_5\), we calculate an estimated mortality rate to compare with actual observations. This analysis reveals model accuracy and areas needing refinement.