Question

the stem-and-leaf plot below gives sample air fares from st. louis to two cities: denver and cincinnati. there were 15 air fares recorded for the denver route and 16 for the cincinnati route. use the plot to answer the questions.
air fare (in dollars)
denver cincinnati
8 6 4 3 2 | 20 | 1 1 6
8 7 3 2 2 | 21 | 1 2 5 5 6 9 9 9
8 8 1 | 22 | 1 2 5
5 | 23 | 1
8 | 24 | 0
(a) which route had more air fares from $220 to $229?
○denver ○cincinnati ○each had the same
(b) which route had the greater median air fare?
○denver ○cincinnati ○the medians were the same
(c) what were the ranges of air fares for the two routes?
denver $□
cincinnati $□

Explanation:

Part (a)

Step1: Count Denver's $220 - $229 fares

Denver's stem - leaf for 22 (representing 220 - 229) has leaves 8, 8, 1. So count is 3.

Step2: Count Cincinnati's $220 - $229 fares

Cincinnati's stem - leaf for 22 (representing 220 - 229) has leaves 1, 2, 5. So count is 3? Wait, no, wait. Wait, Denver's stem 22: leaves 8,8,1 (three values). Cincinnati's stem 22: leaves 1,2,5 (three values? Wait, no, maybe I misread. Wait, the stem - leaf plot: for Denver, stem 22: 8,8,1 (so three fares: 228, 228, 221). For Cincinnati, stem 22: 1,2,5 (221, 222, 225). Wait, but the question is which route had more from $220 - $229. Wait, maybe I made a mistake. Wait, Denver's stem 22: three values, Cincinnati's stem 22: three? Wait, no, maybe the stem for Denver: 22 has leaves 8,8,1 (three), Cincinnati's stem 22 has leaves 1,2,5 (three). But wait, maybe I misread the plot. Wait, the original plot: Denver has 22 | 8 8 1 (so three), Cincinnati has 22 | 1 2 5 (three). Wait, but the options are Denver, Cincinnati, same. Wait, maybe I made a mistake. Wait, no, maybe the stem for Denver: 22 has three, Cincinnati has three? But that can't be. Wait, maybe the stem - leaf for Denver: 22 | 8 8 1 (three), Cincinnati: 22 | 1 2 5 (three). So each had the same? Wait, no, maybe I miscounted. Wait, Denver's stem 22: 8,8,1 (three), Cincinnati's stem 22: 1,2,5 (three). So the answer is Each had the same? Wait, no, maybe I made a mistake. Wait, let's check again. Denver: 220 - 229: the stem is 22, leaves are 8,8,1. So three fares. Cincinnati: stem 22, leaves 1,2,5. Three fares. So each had the same.

Wait, but maybe I misread the plot. Wait, the original plot: Denver has 22 | 8 8 1 (three), Cincinnati has 22 | 1 2 5 (three). So the answer for (a) is Each had the same.

Part (b)

Step1: Find median for Denver (n = 15)

Median is the 8th value (since n = 15, (15 + 1)/2 = 8th term). Let's list Denver's fares:

Stem 20: 202, 203, 204, 206, 208 (wait, stem 20: leaves 2,3,4,6,8? Wait, original plot: Denver: 20 | 2 3 4 6 8 (so 202, 203, 204, 206, 208) - 5 values.

Stem 21: 212, 212, 213, 217, 218 (leaves 2,2,3,7,8) - 5 values. Now we have 5 + 5 = 10 values.

Stem 22: 221, 228, 228 (leaves 1,8,8) - 3 values. Now 10+3 = 13.

Stem 23: 235 (leaf 5) - 1 value. 14.

Stem 24: 248 (leaf 8) - 1 value. 15.

Wait, no, the order should be ascending. Let's list all Denver fares:

202, 203, 204, 206, 208, 212, 212, 213, 217, 218, 221, 228, 228, 235, 248.

The 8th term is 217? Wait, no, 5 (stem 20) + 3 (stem 21: first three? Wait, stem 21 leaves: 2,2,3,7,8. So the values are 212, 212, 213, 217, 218. So after stem 20 (5 values: 202,203,204,206,208), the next 5 are 212,212,213,217,218. So the 8th term is the 3rd in stem 21? Wait, 5 (stem 20) + 3 (212,212,213) = 8th term. So 213? Wait, no, 5 + 3 = 8. So the 8th term is 213?

Now Cincinnati: n = 16, median is the average of 8th and 9th terms.

Cincinnati's fares:

Stem 20: 201, 201, 206 (leaves 1,1,6) - 3 values.

Stem 21: 211, 212, 212, 215, 215, 216, 219, 219 (leaves 1,2,2,5,5,6,9,9) - 8 values. Now 3 + 8 = 11 values.

Stem 22: 221, 222, 225 (leaves 1,2,5) - 3 values. 11 + 3 = 14.

Stem 23: 231 (leaf 1) - 1 value. 15.

Stem 24: 240 (leaf 0) - 1 value. 16.

Wait, list Cincinnati's fares in order:

201, 201, 206, 211, 212, 212, 215, 215, 216, 219, 219, 221, 222, 225, 231, 240.

8th term: 215, 9th term: 216. Median is (215 + 216)/2 = 215.5.

Denver's median: n = 15, 8th term. Let's re - list Denver's fares correctly.

Denver's stem - leaf:

20 | 2 3 4 6 8 → 202, 203, 204, 206, 208 (5)

21 | 2 2 3 7 8 → 212, 212, 213, 217, 218 (5) → total 10

22 | 1 8 8 → 221, 228, 228 (3) → total 13

23 | 5 → 235 (1) → 14

24 | 8 → 248 (1) → 15

So the 8th term is the 3rd term in the stem 21 group? Wait, 5 (stem 20) + 3 (212,212,213) = 8th term. So 213.

Cincinnati's median: (215 + 216)/2 = 215.5. So Denver's median (213) is less than Cincinnati's (215.5). Wait, no, maybe I made a mistake in listing. Wait, Cincinnati's stem 21: leaves 1,2,2,5,5,6,9,9. So the values are 211, 212, 212, 215, 215, 216, 219, 219. So the 8th term is 215, 9th term is 216. Median is 215.5. Denver's 8th term: after 5 (stem 20) and 3 (stem 21: 212,212,213), so 213. So Cincinnati has a greater median.

Part (c)

Step1: Range for Denver

Range = Maximum - Minimum. Denver's minimum: 202 (from stem 20, leaf 2), maximum: 248 (stem 24, leaf 8). So range = 248 - 202 = 46.

Step2: Range for Cincinnati

Cincinnati's minimum: 201 (stem 20, leaf 1), maximum: 240 (stem 24, leaf 0). So range = 240 - 201 = 39.

(a) Answer: Each had the same

(b) Answer: Cincinnati

(c) Denver: $\boldsymbol{46}$, Cincinnati: $\boldsymbol{39}$

Final Answers:

(a) Each had the same

(b) Cincinnati

(c) Denver: 46, Cincinnati: 39

Answer:

Step1: Range for Denver

Range = Maximum - Minimum. Denver's minimum: 202 (from stem 20, leaf 2), maximum: 248 (stem 24, leaf 8). So range = 248 - 202 = 46.

Step2: Range for Cincinnati

Cincinnati's minimum: 201 (stem 20, leaf 1), maximum: 240 (stem 24, leaf 0). So range = 240 - 201 = 39.

(a) Answer: Each had the same

(b) Answer: Cincinnati

(c) Denver: $\boldsymbol{46}$, Cincinnati: $\boldsymbol{39}$

Final Answers:

(a) Each had the same

(b) Cincinnati

(c) Denver: 46, Cincinnati: 39