Question

the stem-and-leaf plot below gives sample air fares from chicago to two cities: boise and detroit. there were 20 air fares recorded for the boise route and 17 for the detroit route. use the plot to answer the questions.
air fare (in dollars)
boise detroit
9 7 1 1 | 20 | 5 9
9 9 5 5 3 2 1 0 | 21 | 0 2 2 2 5 9
9 8 7 6 6 4 | 22 | 1 1 4 5 6 8
4 0 | 23 | 7 7 8
(a) which route had the greater median air fare?
boise detroit the medians were the same
(b) what were the ranges of air fares for the two routes?
boise $□$
detroit $□$
(c) which route had more air fares from $200 to $209?
boise detroit each had the same

Explanation:

Part (a)

Step1: Find median for Boise

Boise has 20 data points. Median is average of 10th and 11th values. Sorting Boise's leaves (for stems 20,21,22,23):
Stem 20: 1,1,7,9 (4 values)
Stem 21: 0,1,2,3,5,5,9,9 (8 values, cumulative 12)
10th value: 21 (from stem 21, 10 - 4 = 6th in stem 21: 0,1,2,3,5,5 → 5)
11th value: 21 (7th in stem 21: 9? Wait, stem 21 leaves: 0,1,2,3,5,5,9,9? Wait, original Boise stem - leaf:
20: 1,1,7,9 (4)
21: 0,1,2,3,5,5,9,9 (8, total 12)
22: 4,6,6,7,8,9 (6, total 18)
23: 0,4 (2, total 20)
Wait, 10th value: index 9 (0 - based) → stem 21, 9 - 4 = 5th (0,1,2,3,5,5 → 5th is 5)
11th value: index 10 → stem 21, 10 - 4 = 6th (9)
Median: (215 + 219)/2 = 217? Wait no, stem is tens place? Wait stem - leaf: stem is 20 (so 200s), leaf is units. So stem 20: 201,201,207,209; stem 21: 210,211,212,213,215,215,219,219; stem 22: 224,226,226,227,228,229; stem 23: 230,234.
So 20 data points. Median is average of 10th and 11th.
1st - 4th: 201,201,207,209
5th - 12th: 210,211,212,213,215,215,219,219 (5th is 210, 12th is 219)
10th value: 10th (index 9) → 215 (9 - 4 = 5th in stem 21: 210(5),211(6),212(7),213(8),215(9),215(10)? Wait no, 0 - based index:
Index 0:201, 1:201, 2:207, 3:209,
4:210, 5:211, 6:212, 7:213, 8:215, 9:215, 10:219, 11:219,
12:224, 13:226, 14:226, 15:227, 16:228, 17:229,
18:230, 19:234
So 10th (index 9): 215, 11th (index 10):219. Median: (215 + 219)/2 = 217.

Step2: Find median for Detroit

Detroit has 17 data points. Median is 9th value (0 - based index 8).
Stem 20: 5,9 (2 values)
Stem 21: 0,2,2,2,5,9 (6 values, cumulative 8)
Stem 22: 1,1,4,5,6,8 (6 values, cumulative 14)
Stem 23: 7,7,8 (3 values, cumulative 17)
Index 8: stem 21, 8 - 2 = 6th? Wait stem 20: 205,209 (indices 0,1)
Stem 21: 210(2),212(3),212(4),212(5),215(6),219(7) → indices 2 - 7
Index 8: stem 22, first value: 221 (index 8)
Wait no, 17 data points: median is (17 + 1)/2 = 9th value (1 - based). So 1 - based index 9:
1 - 2:205,209
3 - 8:210,212,212,212,215,219 (6 values, 2 + 6 = 8)
9th: stem 22, first value:221
So median is 221.

Compare Boise (217) and Detroit (221). Detroit has greater median.

Step1: Range for Boise

Range = Max - Min. Boise: Min = 201, Max = 234. Range = 234 - 201 = 33.

Step2: Range for Detroit

Detroit: Min = 205, Max = 238. Range = 238 - 205 = 33? Wait no:
Boise's min: 201 (stem 20, leaf 1), max:234 (stem 23, leaf 4). 234 - 201 = 33.
Detroit's min:205 (stem 20, leaf 5), max:238 (stem 23, leaf 8). 238 - 205 = 33? Wait 238 - 205 = 33? 238 - 205 = 33? 205 + 33 = 238. Yes. Wait no, 238 - 205 = 33? 238 - 200 = 38, 38 - 5 = 33. Yes. Wait Boise: 234 - 201 = 33. Detroit:238 - 205 = 33? Wait no, 238 - 205 = 33? Wait 205 to 238: 238 - 205 = 33. Wait Boise: 201 to 234: 234 - 201 = 33. Wait is that correct?

Wait Boise's data:
Stem 20: 201,201,207,209 (min 201)
Stem 23: 230,234 (max 234)
234 - 201 = 33.

Detroit's data:
Stem 20:205,209 (min 205)
Stem 23:237,237,238 (max 238)
238 - 205 = 33? Wait 238 - 205 = 33? 205 + 33 = 238. Yes. Wait 238 - 205 = 33. So both ranges? Wait no, maybe I made a mistake. Wait Boise's max is 234, min 201: 234 - 201 = 33. Detroit's max 238, min 205: 238 - 205 = 33. Wait that's same? Wait no, 238 - 205 = 33? 205 + 33 = 238. Yes. So Boise range: 33, Detroit range: 33? Wait no, 238 - 205 = 33? Wait 238 - 200 = 38, 38 - 5 = 33. Yes. So both ranges are 33? Wait maybe I miscalculated. Wait Boise: 234 - 201 = 33. Detroit:238 - 205 = 33. So both ranges are 33.

Wait no, let's check again:

Boise:
Min: 201 (from stem 20, leaf 1)
Max: 234 (from stem 23, leaf 4)
Range = 234 - 201 = 33.

Detroit:
Min: 205 (from stem 20, leaf 5)
Max: 238 (from stem 23, leaf 8)
Range = 238 - 205 = 33.

Yes, both ranges are 33.

Step1: Count Boise's $200 - 209$

Boise's stem 20: leaves 1,1,7,9 → 4 values (201,201,207,209)

Step2: Count Detroit's $200 - 209$

Detroit's stem 20: leaves 5,9 → 2 values (205,209)

Boise has 4, Detroit has 2. So Boise has more.

Answer:

Detroit

Part (b)