Provide a complete picture

Oct 13, 2025 by recoveryshake.com 11 views

The Data Analysis of Licensed Drivers by Age and Sex

The data in the table represent the number of licensed drivers by age and sex. In this analysis, we will focus on finding the least-squares regression line for males, using the number of fatal crashes, ${ y }$ , as the dependent variable, and the age of the drivers as the independent variable. This analysis will help us understand the relationship between the age of male drivers and the number of fatal crashes.

Age	Number of Licensed Drivers (Males)	Number of Fatal Crashes
16-20	1000	50
21-25	1200	60
26-30	1500	70
31-35	1800	80
36-40	2000	90
41-45	2200	100
46-50	2500	110
51-55	2800	120
56-60	3000	130
61-65	3200	140
66-70	3500	150

Part (a) - Finding the Least-Squares Regression Line

To find the least-squares regression line, we need to calculate the slope and the intercept of the line. The slope of the line represents the change in the dependent variable (number of fatal crashes) for a one-unit change in the independent variable (age of the drivers). The intercept of the line represents the value of the dependent variable when the independent variable is equal to zero.

The formula for the slope of the least-squares regression line is:

${ \beta_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} }$

where ${ x_i }$ is the age of the ${ i }$ th driver, ${ y_i }$ is the number of fatal crashes for the ${ i }$ th driver, ${ \bar{x} }$ is the mean age of the drivers, and ${ \bar{y} }$ is the mean number of fatal crashes.

The formula for the intercept of the least-squares regression line is:

${ \beta_0 = \bar{y} - \beta_1 \bar{x} }$

Using the data in the table, we can calculate the slope and the intercept of the least-squares regression line.

Calculating the Slope

First, we need to calculate the mean age of the drivers and the mean number of fatal crashes.

${ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{16+21+26+31+36+41+46+51+56+61+66}{11} = 38.55 }$

${ \bar{y} = \frac{\sum_{i=1}^{n} y_i}{n} = \frac{50+60+70+80+90+100+110+120+130+140+150}{11} = 93.64 }$

Next, we need to calculate the sum of the products of the deviations of the age and the number of fatal crashes from their means.

${ \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = (16-38.55)(50-93.64) + (21-38.55)(60-93.64) + (26-38.55)(70-93.64) + (31-38.55)(80-93.64) + (36-38.55)(90-93.64) + (41-38.55)(100-93.64) + (46-38.55)(110-93.64) + (51-38.55)(120-93.64) + (56-38.55)(130-93.64) + (61-38.55)(140-93.64) + (66-38.55)(150-93.64) }$

${ = -22.55 \times -43.64 + -17.55 \times -33.64 + -12.55 \times -23.64 + -7.55 \times -13.64 + -2.55 \times -3.64 + 2.45 \times 6.36 + 7.45 \times 16.36 + 12.45 \times 26.36 + 17.45 \times 36.36 + 22.45 \times 46.36 + 27.45 \times 56.36 }$

${ = 981.19 + 588.19 + 294.19 + 102.19 + 9.19 + 15.49 + 121.19 + 326.19 + 632.19 + 1039.19 + 1569.19 }$

${ = 6618.19 }$

Finally, we can calculate the slope of the least-squares regression line.

${ \beta_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} = \frac{6618.19}{\sum_{i=1}^{n} (x_i - 38.55)^2} }$

To calculate the denominator, we need to calculate the sum of the squared deviations of the age from the mean age.

${ \sum_{i=1}^{n} (x_i - \bar{x})^2 = (16-38.55)^2 + (21-38.55)^2 + (26-38.55)^2 + (31-38.55)^2 + (36-38.55)^2 + (41-38.55)^2 + (46-38.55)^2 + (51-38.55)^2 + (56-38.55)^2 + (61-38.55)^2 + (66-38.55)^2 }$

${ = 22.55^2 + 17.55^2 + 12.55^2 + 7.55^2 + 2.55^2 + 2.45^2 + 7.45^2 + 12.45^2 + 17.45^2 + 22.45^2 + 27.45^2 }$

${ = 509.19 + 307.19 + 156.19 + 56.19 + 6.49 + 6.00 + 55.19 + 155.19 + 304.19 + 503.19 + 753.19 }$

${ = 2803.19 }$

Now, we can calculate the slope of the least-squares regression line.

${ \beta_1 = \frac{6618.19}{2803.19} = 2.36 }$

Calculating the Intercept

Now that we have calculated the slope of the least-squares regression line, we can calculate the intercept of the line.

${ \beta_0 = \bar{y} - \beta_1 \bar{x} = 93.64 - 2.36 \times 38.55 = 93.64 - 90.93 = 2.71 }$

The Least-Squares Regression Line

The least-squares regression line for males, using the number of fatal crashes, ${ y }$ , as the dependent variable, and the age of the drivers as the independent variable, is:

${ y = 2.71 + 2.36x }$

This line represents the relationship between the age of male drivers and the number of fatal crashes.

In this analysis, we have found the least-squares regression line for males, using the number of fatal crashes, ${ y }$ , as the dependent variable, and the age of the drivers as the independent variable. The line is:

${ y = 2.71 + 2.36x }$

This line represents the relationship between the age of male drivers and the number of fatal crashes. The slope of the line is 2.36, which means that for every one-unit increase in the age of the drivers, the number of fatal crashes increases by 2.36 units. The intercept of the line is 2.71, which means that when the age of the drivers is equal to zero, the number of fatal crashes is equal to 2.71 units.

Part (b) - Interpreting the Results

The results of this analysis can be interpreted in several ways. First, the positive slope of the line indicates that there is a positive relationship between the age of male drivers and the number of fatal crashes. This means that as the age of the drivers increases, the number of fatal crashes also increases.

Second, the magnitude of the slope indicates that the relationship between the age of male drivers and the number of fatal crashes is strong. The slope of 2.36 indicates that for every one-unit increase in the age of the drivers, the number of fatal crashes increases by 2.36 units.

Third, the intercept of the line indicates that when the age of the drivers is equal to zero, the number of fatal crashes is equal to 2.71 units.
Q&A: The Data Analysis of Licensed Drivers by Age and Sex

In our previous article, we analyzed the data on licensed drivers by age and sex, and found the least-squares regression line for males, using the number of fatal crashes, ${ y }$ , as the dependent variable, and the age of the drivers as the independent variable. In this article, we will answer some frequently asked questions (FAQs) related to the analysis.

Q: What is the purpose of the analysis?

A: The purpose of the analysis is to understand the relationship between the age of male drivers and the number of fatal crashes.

Q: What is the least-squares regression line?

A: The least-squares regression line is a mathematical equation that represents the relationship between the independent variable (age of the drivers) and the dependent variable (number of fatal crashes).

Q: What is the slope of the least-squares regression line?

A: The slope of the least-squares regression line is 2.36, which means that for every one-unit increase in the age of the drivers, the number of fatal crashes increases by 2.36 units.

Q: What is the intercept of the least-squares regression line?

A: The intercept of the least-squares regression line is 2.71, which means that when the age of the drivers is equal to zero, the number of fatal crashes is equal to 2.71 units.

Q: What does the positive slope of the line indicate?

A: The positive slope of the line indicates that there is a positive relationship between the age of male drivers and the number of fatal crashes. This means that as the age of the drivers increases, the number of fatal crashes also increases.

Q: What does the magnitude of the slope indicate?

A: The magnitude of the slope indicates that the relationship between the age of male drivers and the number of fatal crashes is strong. The slope of 2.36 indicates that for every one-unit increase in the age of the drivers, the number of fatal crashes increases by 2.36 units.

Q: What does the intercept of the line indicate?

A: The intercept of the line indicates that when the age of the drivers is equal to zero, the number of fatal crashes is equal to 2.71 units.

Q: What are the implications of the analysis?

A: The analysis has several implications. First, it suggests that older drivers are more likely to be involved in fatal crashes. Second, it suggests that the relationship between age and fatal crashes is strong. Finally, it suggests that policymakers and safety advocates should consider the age of drivers when developing strategies to reduce the number of fatal crashes.

Q: What are the limitations of the analysis?

A: The analysis has several limitations. First, it is based on a small sample size. Second, it does not control for other factors that may influence the relationship between age and fatal crashes. Finally, it does not provide a complete picture of the relationship between age and fatal crashes.

In this article, we have answered some frequently asked questions (FAQs) related to the analysis of the data on licensed drivers by age and sex. The analysis suggests that there is a positive relationship between the age of male drivers and the number of fatal crashes. The slope of the least-squares regression line is 2.36, which means that for every one-unit increase in the age of the drivers, the number of fatal crashes increases by 2.36 units. The intercept of the line is 2.71, which means that when the age of the drivers is equal to zero, the number of fatal crashes is equal to 2.71 units.

Future research should aim to:

Increase the sample size: The analysis is based on a small sample size. Future research should aim to increase the sample size to provide a more complete picture of the relationship between age and fatal crashes.
Control for other factors: The analysis does not control for other factors that may influence the relationship between age and fatal crashes. Future research should aim to control for these factors to provide a more accurate picture of the relationship.
Provide a complete picture: The analysis does not provide a complete picture of the relationship between age and fatal crashes. Future research should aim to provide a complete picture of the relationship by considering other factors that may influence the relationship.

By addressing these limitations, future research can provide a more accurate and complete picture of the relationship between age and fatal crashes.