Question

a study was conducted in 2023 to try and determine the average commute time for americans. this data was based on commute times from across the nation. a researcher wanted to see if this data accurately reflected the commute times for drivers in her county so she decided to randomly sample 3000 workers from her county and record their commute times. the data is presented in the table below. use this information to answer this question.

commute times

commute times	frequency	relative frequency	cumulative frequency	cumulative relative frequency
less than 15 minutes	a	32%	c	59.5%
15 - 29 minutes	615	20.5%	2400	d
30 - 59 minutes	495	b	2895	96.5%
60 - 119 minutes	75	2.5%	2970	99%
120 minutes or more	30	1%	3000	100%

based on the table above, identify the four missing values of a, b, c, and d. make sure clearly label which number corresponds to each letter and make sure to show your work as in how you arrived at your answer. simply writing the answers will not earn full credit on this problem.

Explanation:

for A:

Step1: Recall relative frequency formula

Relative Frequency = $\frac{\text{Frequency}}{\text{Total Number of Observations}}$. Total observations = 3000. So, Frequency (A) = Relative Frequency × Total.
$A = 0.32\times3000 = 960$

Step2: Find Cumulative Frequency (C)

Cumulative Frequency is sum of previous frequencies. Previous frequency (No Commute) is 825, so $C = 825 + A = 825 + 960 = 1785$

Step3: Calculate Relative Frequency (B)

Relative Frequency (B) = $\frac{\text{Frequency of 30 - 59 Minutes}}{\text{Total}} = \frac{495}{3000} = 0.165 = 16.5\%$

Step4: Find Cumulative Relative Frequency (D)

Cumulative Relative Frequency is sum of previous relative frequencies. Previous cumulative relative frequency (after 15 - 29) is 59.5%, add 20.5%: $D = 59.5\% + 20.5\% = 80\%$

for B:

Step1: Recall relative frequency formula

Relative Frequency = $\frac{\text{Frequency}}{\text{Total Number of Observations}}$. Total observations = 3000. So, Frequency (A) = Relative Frequency × Total.
$A = 0.32\times3000 = 960$

Step2: Find Cumulative Frequency (C)

Cumulative Frequency is sum of previous frequencies. Previous frequency (No Commute) is 825, so $C = 825 + A = 825 + 960 = 1785$

Step3: Calculate Relative Frequency (B)

Relative Frequency (B) = $\frac{\text{Frequency of 30 - 59 Minutes}}{\text{Total}} = \frac{495}{3000} = 0.165 = 16.5\%$

Step4: Find Cumulative Relative Frequency (D)

Cumulative Relative Frequency is sum of previous relative frequencies. Previous cumulative relative frequency (after 15 - 29) is 59.5%, add 20.5%: $D = 59.5\% + 20.5\% = 80\%$

Answer:

• \( A = 960 \) (Frequency for "Less than 15 Minutes", calculated as \( 0.32 \times 3000 \))
• \( B = 16.5\% \) (Relative Frequency for "30 - 59 Minutes", calculated as \( \frac{495}{3000} \))
• \( C = 1785 \) (Cumulative Frequency, sum of 825 and 960)
• \( D = 80\% \) (Cumulative Relative Frequency, sum of 59.5% and 20.5%)