Question

The average number of minutes Americans commute to work is 27.7 minutes. The average commute time in minutes for 48 cities are as follows. Albuquerque 23.6 Jacksonville 26.5 Phoenix 28.6 Atlanta 28.6 Kansas City 23.7 Pittsburgh 25.3 Austin 24.9 Las Vegas 28.7 Portland 26.7 Baltimore 32.4 Little Rock 20.4 Providence 23.9 Boston 32.0 Los Angeles 32.5 Richmond 23.7 Charlotte 26.1 Louisville 21.7 Sacramento 26.1 Chicago 38.4 Memphis 24.1 Salt Lake City 20.5 Cincinnati 25.2 Miami 31.0 San Antonio 26.4 Cleveland 27.1 Milwaukee 25.1 San Diego 25.1 Columbus 23.7 Minneapolis 23.9 San Francisco 32.9 Dallas 28.8 Nashville 25.6 San Jose 28.8 Denver 28.4 New Orleans 32.0 Seattle 27.6 Detroit 29.6 New York 44.1 St. Louis 27.1 El Paso 24.7 Oklahoma City 22.3 Tucson 24.3 Fresno 23.3 Orlando 27.4 Tulsa 20.4 Indianapolis 25.1 Philadelphia 34.5 Washington, D.C. 33.1 (a) What is the mean commute time (in minutes) for these 48 cities? (Round your answer to one decimal place.) minutes (b) Compute the median commute time (in minutes). minutes (c) Compute the mode(s) (in minutes). (Enter your answers as a comma-separated list.)

Answer 1

Answer:

a) $\bar X = \frac{\sum_{i=1}^n X_i}{n}$

And replacing we got:

$\bar X = 27.2$

b) For this case we have n =48 observations and we can calculate the median with the average between the 24th and 25th values on the dataset ordered.

20.4 20.4 20.5 21.7 22.3 23.3 23.6 23.7 23.7 23.7  23.9 23.9 24.1 24.3 24.7 24.9 25.1 25.1 25.1 25.2  25.3 25.6 26.1 26.1 26.4 26.5 26.7 27.1 27.1 27.4  27.6 28.4 28.6 28.6 28.7 28.8 28.8 29.6 31.0 32.0  32.0 32.4 32.5 32.9 33.1 34.5 38.4 44.1

For this case the median would be:

$Median = \frac{26.1+26.4}{2}=26.25 \approx 26.3$

c) $Mode= 23.2, 25.1$

And both with a frequency of 3 so then we have a bimodal distribution for this case

Step-by-step explanation:

For this case we have the following dataset:

23.6, 26.5, 28.6, 28.6, 23.7, 25.3, 24.9, 28.7, 26.7, 32.4, 20.4, 23.9, 32.0, 32.5, 23.7, 26.1, 21.7, 26.1, 38.4, 24.1, 20.5, 25.2, 31, 26.4, 27.1 ,25.1, 25.1, 23.7, 23.9, 32.9, 28.8, 25.6, 28.8, 28.4, 32, 27.6, 29.6, 44.1, 27.1, 24.7, 22.3, 24.3, 23.3, 27.4, 20.4, 25.1, 34.5, 33.1

Part a

We can calculate the mean with the following formula:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

And replacing we got:

$\bar X = 27.2$

Part b

For this case we have n =48 observations and we can calculate the median with the average between the 24th and 25th values on the dataset ordered.

20.4 20.4 20.5 21.7 22.3 23.3 23.6 23.7 23.7 23.7  23.9 23.9 24.1 24.3 24.7 24.9 25.1 25.1 25.1 25.2  25.3 25.6 26.1 26.1 26.4 26.5 26.7 27.1 27.1 27.4  27.6 28.4 28.6 28.6 28.7 28.8 28.8 29.6 31.0 32.0  32.0 32.4 32.5 32.9 33.1 34.5 38.4 44.1

For this case the median would be:

$Median = \frac{26.1+26.4}{2}=26.25 \approx 26.3$

Part c

For this case the mode would be:

$Mode= 23.2, 25.1$

And both with a frequency of 3 so then we have a bimodal distribution for this case