Question

1. forty students were asked to report the time of their commute to school, in minutes. the data are shown in the dot - plot

a. the mean commute time to school is 14.2 minutes. how does the median commute time compare to this value? explain your reasoning.

b. which measure of center, mean or median, would best describe the typical commute time to school? explain.

c. the standard deviation of commute times for these 40 students is 14.4 minutes. interpret this value.

Explanation:

Step1: Find the median position

Since there are $n = 40$ data - points, the median is the average of the 20th and 21st ordered data - points. Looking at the dot - plot, we can count the number of data - points to find these values.

Step2: Analyze the distribution

The distribution appears to be skewed to the right (there are some large values pulling the mean up). In a right - skewed distribution, the mean is typically greater than the median.

Step3: Answer part a

The median is likely less than the mean of 14.2 minutes. This is because the right - skewed distribution has some large values that increase the mean but have less impact on the median.

Step4: Answer part b

The median would best describe the typical commute time. In a skewed distribution, the median is less affected by extreme values. The mean is pulled in the direction of the extreme values (in this case, the long commute times), so it may not represent the "typical" commute as well as the median.

Step5: Answer part c

The standard deviation of 14.4 minutes represents the average amount by which the commute times of the 40 students differ from the mean commute time of 14.2 minutes. A value of 14.4 minutes indicates that there is a relatively large spread in the commute times of the students. Some students have commute times that are close to the mean, while others have commute times that are quite a bit different from the mean.

Answer:

a. The median is likely less than 14.2 minutes because the distribution is right - skewed and extreme values pull the mean up.
b. The median would best describe the typical commute time as it is less affected by extreme values in the right - skewed distribution.
c. The standard deviation of 14.4 minutes represents the average amount by which the commute times of the 40 students differ from the mean commute time. It indicates a relatively large spread in the commute times.