Question

The following table shows the number of licensed U.S. drivers by age and by sex (www.dot.gov): $$ \begin{array}{l|rr|r} \text { Age } & \begin{array}{c} \text { Male Drivers } \\ \text { (number) } \end{array} & \begin{array}{c} \text { Female Drivers } \\ \text { (number) } \end{array} & \text { Total } \\ \hline 19 \text { and under } & 4,777,694 & 4,553,946 & \mathbf{9 , 3 3 1 , 6 4 0} \\ 20-24 & 8,611,161 & 8,398,879 & \mathbf{1 7 , 0 1 0 , 0 4 0} \\ 25-29 & 8,879,476 & 8,666,701 & \mathbf{1 7 , 5 4 6 , 1 7 7} \\ 30-34 & 9,262,713 & 8,997,662 & \mathbf{1 8 , 2 6 0 , 3 7 5} \\ 35-39 & 9,848,050 & 9,576,301 & \mathbf{1 9 , 4 2 4 , 3 5 1} \\ 40-44 & 10,617,456 & 10,484,149 & \mathbf{2 1 , 1 0 1 , 6 0 5} \\ 45-49 & 10,492,876 & 10,482,479 & \mathbf{2 0 , 9 7 5 , 3 5 5} \\ 50-54 & 9,420,619 & 9,475,882 & \mathbf{1 8 , 8 9 6 , 5 0 1} \\ 55-59 & 8,218,264 & 8,265,775 & \mathbf{1 6 , 4 8 4 , 0 3 9} \\ 60-64 & 6,103,732 & 6,147,569 & \mathbf{1 2 , 2 5 1 , 3 6 1} \\ 65-69 & 4,571,157 & 4,643,913 & \mathbf{9 , 2 1 5 , 0 7 0} \\ 70-74 & 3,617,908 & 3,761,039 & \mathbf{7 , 3 7 8 , 9 4 7} \\ 75-79 & 2,890,155 & 3,192,408 & \mathbf{6 , 0 8 2 , 5 6 3} \\ 80-84 & 1,907,743 & 2,222,412 & \mathbf{4 , 1 3 0 , 1 5 5} \\ 85 \text { and over } & 1,170,817 & 1,406,271 & \mathbf{2 , 5 7 7 , 0 8 8} \\ \hline \text { Total } & \mathbf{1 0 0 , 3 8 9 , 8 8 1} & \mathbf{1 0 0 , 2 7 5 , 3 8 6} & \mathbf{2 0 0 , 6 6 5 , 2 6 7} \end{array} $$ a) What percent of total drivers are under 20 ? b) What percent of total drivers are male? c) Write a few sentences comparing the number of male and female licensed drivers in each age group. d) Do a driver's age and sex appear to be independent? Explain?

Short answer

a) 4.65% of drivers are under 20. b) 50.05% of drivers are male. c) Younger and older age groups have more female drivers; mid-ages often have more males. d) Age and sex likely depend, suggested by varied driver numbers.

Step-by-step solution

Calculate Percentage of Drivers Under 20

To find the percentage of total drivers who are under 20, use the formula \( \frac{\text{Number of drivers under 20}}{\text{Total drivers}} \times 100 \). Here, the number of drivers under 20 is 9,331,640, and the total number is 200,665,267. Thus, the percentage is \( \frac{9,331,640}{200,665,267} \times 100 \approx 4.65\% \).

Calculate Percentage of Male Drivers

To calculate the percentage of drivers who are male, use the formula \( \frac{\text{Number of male drivers}}{\text{Total drivers}} \times 100 \). Here, the total number of male drivers is 100,389,881. Hence, the percentage is \( \frac{100,389,881}{200,665,267} \times 100 \approx 50.05\% \).

Compare Male and Female Drivers in Each Age Group

By examining each age group's data from the table, we see that generally, for younger ages (19 and under), and older ages (80 and over), there are more female drivers than male. In mid-age groups like 40-44, males slightly outnumber females. These variations could reflect demographic trends or differences in licensing behaviors between genders.

Assess Independence of Age and Sex

For age and sex to be independent, the likelihood of drivers being male or female should be constant across all age groups. Observing the data, the variations in numbers of male versus female drivers in different age categories seem substantial, hinting that age and sex might not be independent. Statistical tests can confirm this hypothesis, but the data suggests a dependency due to noticeable patterns and differences in driver numbers for certain age groups.

Key concepts

Understanding Percentages

Percentages are a way to express numbers as a fraction of 100, making it easier to understand proportions. In statistical analysis, percentages help us to grasp the relative share of a subgroup within a total population.
For instance, in the problem here, we calculate the percentage of drivers under 20. We do this by dividing the number of drivers in this category by the total number of drivers. So, we perform the division \( \frac{9,331,640}{200,665,267} \), which gives us a fraction.
To convert this fraction to a percentage, we multiply by 100, producing approximately 4.65%. This figure tells us that drivers under 20 represent about 4.65% of all licensed drivers in the data set, giving us an insightful glimpse into the age distribution of drivers.

Comparative Analysis Between Male and Female Drivers

Comparative analysis involves examining the similarities and differences between two sets of data. Here, we compare male and female drivers across varying age groups.
Such an analysis helps identify patterns and trends. From the table, we observe that for younger (19 and under) and older age brackets (80 and over), female drivers outnumber male drivers. Conversely, in middle age groups, such as 40-44, male drivers slightly outnumber females.
This could indicate cultural or social preferences in acquiring driver's licenses, or other demographic factors that influence these numbers. Understanding these patterns provides valuable insights into gender trends over different stages of life.

Testing for Independence: Age and Sex

An independence test in statistics checks if two variables are related. In our case, we examine whether a driver's age is independent of their sex. This means we want to see if being male or female has no influence on the driver's age category or vice versa.
If data show that the distribution of males and females is consistent across all age groups, age and sex can be considered independent. However, the given data illustrates significant differences in male and female driver numbers across various age categories.
This might suggest a dependency between age and sex in driving demographics, indicating that one variable can inform us about the other. To conclusively determine independence, statistical tests such as the Chi-square test could be applied, but preliminary data hints at some interdependence.