Question

activity 1: conditional crash rates
the national safety council uses statistical techniques to estimate the number of crashes by age group. the table summarizes information for u.s. licensed drivers in 2022.

	in a crash	not in a crash	total
25+ years old	17,640,000	201,381,000	219,021,000
total	22,820,000	212,266,000	235,086,000

1. divide the value in each cell by the total number of drivers to find each probability to two decimal places. some of the values have been computed for you.

relative frequency table

	in a crash	not in a crash	total
25+ years old			0.93
total			1.00

Answer:

To solve this, we'll calculate the relative frequencies (probabilities) for each cell by dividing the cell value by the total number of drivers (229,086,000).

Step 1: 16–24 years old – In a crash

The total for 16–24 years old is \( 26,905,000 \). The "Not in a crash" value is \( 20,885,000 \), so "In a crash" is \( 26,905,000 - 20,885,000 = 6,020,000 \) (or directly use the given \( 5,180,000 \)? Wait, the first table shows 16–24: In a crash = 5,180,000; Not in a crash = 20,885,000; Total = 26,905,000. Let's use that.

Probability = \( \frac{5,180,000}{229,086,000} \approx 0.02 \) (to two decimals).

Step 2: 16–24 years old – Total

Total for 16–24 is \( 26,905,000 \). Probability = \( \frac{26,905,000}{229,086,000} \approx 0.12 \) (to two decimals).

Step 3: 25+ years old – In a crash

Total for 25+ is \( 202,181,000 \) (from first table: 17,640,000 + 184,541,000? Wait, first table: 25+ years old: In a crash = 17,640,000; Not in a crash = 184,541,000; Total = 202,181,000. Total drivers = 229,086,000.

Probability = \( \frac{17,640,000}{229,086,000} \approx 0.08 \) (to two decimals).

Step 4: 25+ years old – Not in a crash

Probability = \( \frac{184,541,000}{229,086,000} \approx 0.81 \) (Wait, the given "25+ years old – Total" is 0.89? Wait, no—wait the second table has "25+ years old – Total" as 0.89? Wait, no, let's recheck.

Wait, the total number of drivers is 229,086,000.

For 16–24:

• In a crash: \( \frac{5,180,000}{229,086,000} \approx 0.02 \)
• Not in a crash: \( \frac{20,885,000}{229,086,000} \approx 0.09 \) (matches the given 0.09)
• Total: \( \frac{26,905,000}{229,086,000} \approx 0.12 \)

For 25+ years old:

• Total: \( \frac{202,181,000}{229,086,000} \approx 0.88 \) (but the table says 0.89? Maybe rounding differences. Let's use the given "Total" row: Total drivers = 229,086,000, so total probability is 1.00.

• In a crash: Total "In a crash" is 22,820,000 (5,180,000 + 17,640,000). So \( \frac{22,820,000}{229,086,000} \approx 0.10 \) (sum of 16–24 In a crash (0.02) and 25+ In a crash (0.08) is 0.10, which matches \( \frac{22,820,000}{229,086,000} \approx 0.10 \)).

• Not in a crash: Total "Not in a crash" is 212,261,000 (20,885,000 + 191,376,000? Wait first table: Not in a crash total is 212,261,000. So \( \frac{212,261,000}{229,086,000} \approx 0.93 \)? No, wait the second table has "Total – Not in a crash" as \( 1.00 - 0.10 = 0.90 \)? Maybe my initial numbers are off. Let's use the given partial values:

Given:

• 16–24: Not in a crash = 0.09 (wait, no—wait the second table's "16–24 years old – Not in a crash" is 0.09? Wait, no, the first table's 16–24 Not in a crash is 20,885,000. \( 20,885,000 / 229,086,000 ≈ 0.09 \) (correct). Then 16–24 Total is \( 26,905,000 / 229,086,000 ≈ 0.12 \).

25+ years old Total: \( 1.00 - 0.12 = 0.88 \) (but the table says 0.89? Maybe rounding to two decimals: \( 202,181,000 / 229,086,000 ≈ 0.88 \) or 0.89).

25+ years old In a crash: Total In a crash is 22,820,000. 16–24 In a crash is 5,180,000, so 25+ In a crash is 22,820,000 - 5,180,000 = 17,640,000. \( 17,640,000 / 229,086,000 ≈ 0.08 \).

25+ years old Not in a crash: Total Not in a crash is 212,261,000. 16–24 Not in a crash is 20,885,000, so 25+ Not in a crash is 212,261,000 - 20,885,000 = 191,376,000. \( 191,376,000 / 229,086,000 ≈ 0.83 \) (Wait, 1.00 - 0.12 (16–24 Total) - 0.08 (25+ In a crash) = 0.80? No, I'm getting confused. Let's use the relative frequency table structure:

	In a crash	Not in a crash	Total
25+ years old	\( \approx 0.08 \)	\( \approx 0.81 \)	\( \approx 0.89 \)
Total	\( \approx 0.10 \)	\( \approx 0.90 \)	1.00

Final Relative Frequency Table (rounded to two decimals):

	In a crash	Not in a crash	Total
25+ years old	\( 0.08 \)	\( 0.81 \)	\( 0.89 \)
Total	\( 0.10 \)	\( 0.90 \)	\( 1.00 \)

(Note: Rounding may vary slightly based on calculation precision, but these are the approximate values.)