Question

students at a major university in southern california are complaining of a serious housing crunch. according to many of these students, they have to commute too far to school, and so university officials should build more housing near campus. in response, the officials studied the commute distance (in miles) of 70 students at the university and published the following frequency distribution in the school paper.

commute distance (miles)	1 to 5	6 to 10	11 to 15	16 to 20	21 to 25	26 to 30

based on the frequency distribution, using the midpoint of each data class, estimate the mean commute distance for the students. for your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

Explanation:

Step1: Find midpoints of each class

For class 1 to 5: Midpoint $m_1 = \frac{1 + 5}{2} = 3$
For class 6 to 10: Midpoint $m_2 = \frac{6 + 10}{2} = 8$
For class 11 to 15: Midpoint $m_3 = \frac{11 + 15}{2} = 13$
For class 16 to 20: Midpoint $m_4 = \frac{16 + 20}{2} = 18$
For class 21 to 25: Midpoint $m_5 = \frac{21 + 25}{2} = 23$
For class 26 to 30: Midpoint $m_6 = \frac{26 + 30}{2} = 28$

Step2: Calculate $f \times m$ for each class

• Class 1 - 5: $f_1 = 20$, $f_1 \times m_1 = 20 \times 3 = 60$
• Class 6 - 10: $f_2 = 21$, $f_2 \times m_2 = 21 \times 8 = 168$
• Class 11 - 15: $f_3 = 15$, $f_3 \times m_3 = 15 \times 13 = 195$
• Class 16 - 20: $f_4 = 8$, $f_4 \times m_4 = 8 \times 18 = 144$
• Class 21 - 25: $f_5 = 4$, $f_5 \times m_5 = 4 \times 23 = 92$
• Class 26 - 30: $f_6 = 2$, $f_6 \times m_6 = 2 \times 28 = 56$

Step3: Find total frequency ($N$) and total $f \times m$

Total frequency $N = 20 + 21 + 15 + 8 + 4 + 2 = 70$
Total $f \times m = 60 + 168 + 195 + 144 + 92 + 56 = 715$

Step4: Calculate the mean

Mean $\bar{x} = \frac{\sum (f \times m)}{N} = \frac{715}{70} \approx 10.2$

Answer:

10.2