A researcher found a significant relationship between a​ person's age, 

x 1x1​, 

the number of hours a person works per​ week, 

x 2x2​, 

and the number of​ accidents, y, the person has per year. The relationship can be represented by the multiple regression equation 

ModifyingAbove y with caret equals negative 3.2 plus 0.012 x 1 plus 0.23 x 2 .y=−3.2+0.012x1+0.23x2. 

Predict the number of accidents per year​ (to the nearest whole​ number) for a person whose age is 

4949 

and who works 

5454 

hours per week.

The given multiple regression equation is:

\[
\hat{y} = -3.2 + 0.012x_1 + 0.23x_2
\]

where:
- \(\hat{y}\) represents the predicted number of accidents per year.
- \(x_1\) is the person's age.
- \(x_2\) is the number of hours the person works per week.

We need to predict the number of accidents per year for a person who is 49 years old and works 54 hours per week. Plug the values \(x_1 = 49\) and \(x_2 = 54\) into the equation:

\[
\hat{y} = -3.2 + 0.012(49) + 0.23(54)
\]

Let's calculate this step by step:

1. Calculate \(0.012 \times 49\):
   \[
   0.012 \times 49 = 0.588
   \]

2. Calculate \(0.23 \times 54\):
   \[
   0.23 \times 54 = 12.42
   \]

3. Now, sum these values along with \(-3.2\):
   \[
   \hat{y} = -3.2 + 0.588 + 12.42 = 9.808
   \]

Since the problem asks for the number of accidents per year to the nearest whole number:

\[
\hat{y} \approx 10
\]

So, the predicted number of accidents per year is **10**.

Would you like more details on how the calculation works or have any other questions? Here are some related questions:

1. How would the prediction change if the age were increased to 60 years?
2. What happens to the prediction if the person works fewer hours, say 40 hours per week?
3. Can you explain what each coefficient in the regression equation represents?
4. How can this model be used to assess risk in different work environments?
5. What is the impact of multicollinearity on multiple regression models?
6. How would you calculate the residual for a person with a known number of accidents?
7. Can this regression equation be used to make predictions for extreme values of age or work hours?
8. How does one determine if a regression model is a good fit for the data?

**Tip:** Always check the range of your input variables to ensure they fall within the scope of the data used to create the regression model. This ensures more accurate predictions.

Learn how to predict the number of accidents per year using a multiple regression equation. This problem involves calculating the predicted accidents based on age and hours worked per week. Suitable for advanced learners and researchers.

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