Question

a survey, 16 high school graduates and 19 non-graduates were asked a question:
\how many hours did you spend working at your job last week?\
the results are shown in the stem-and-leaf plot below. use the plot to answer the questions.

number of hours at work
graduates non-graduates
8 7 2 | 2 | 1 3
5 3 2 2 | 3 | 0 2 8 8 8
6 5 4 3 1 0 0 | 4 | 0 3 4 5 6 7 8 8
8 0 | 5 | 0 1 2 2

(a) which group had the greater median number of hours?
○graduates ○non-graduates ○the medians were the same

(b) which group had more responses in the 20s?
○graduates ○non-graduates ○each had the same

(c) what were the ranges for the two groups?
graduates □ hours
non-graduates □ hours

Explanation:

Part (a)

Step1: Find median for Graduates

There are 16 graduates (even number). The median is the average of the 8th and 9th values. Ordering the stem - leaf data for graduates:
Stem 2: 2, 7, 8 (3 values)
Stem 3: 2, 2, 3, 5 (4 values, cumulative: 3 + 4 = 7)
Stem 4: 0, 0, 1, 3, 4, 5, 6 (7 values, cumulative: 7+7 = 14) Wait, no, let's list all graduate data points:
From stem - leaf:
Stem 2: 2, 7, 8 (3)
Stem 3: 2, 2, 3, 5 (4, total so far 3 + 4 = 7)
Stem 4: 0, 0, 1, 3, 4, 5, 6 (7, total so far 7+7 = 14)
Stem 5: 0, 8 (2, total 14 + 2 = 16)
So the 8th value is in stem 4: the 8th value (since first 7 are up to stem 3) is the first value in stem 4? Wait, no. Let's count properly:
1:2, 2:7, 3:8, 4:2, 5:2, 6:3, 7:5, 8:0 (stem 4, first value), 9:0 (stem 4, second value)
So the 8th value is 0 (stem 4) and 9th value is 0 (stem 4)? Wait, no, I think I messed up the stem - leaf reading. The stem - leaf for graduates:
Graduates:
Stem 2: 2, 7, 8 (so data points: 22, 27, 28)
Stem 3: 2, 2, 3, 5 (data points: 32, 32, 33, 35)
Stem 4: 0, 0, 1, 3, 4, 5, 6 (data points: 40, 40, 41, 43, 44, 45, 46)
Stem 5: 0, 8 (data points: 50, 58)
Now, list all 16 data points in order:
22, 27, 28, 32, 32, 33, 35, 40, 40, 41, 43, 44, 45, 46, 50, 58
The 8th value is 40, the 9th value is 40. Median for graduates: $\frac{40 + 40}{2}=40$

Step2: Find median for Non - graduates

There are 19 non - graduates (odd number). The median is the 10th value.
Non - graduate data points:
Stem 2: 1, 3 (2)
Stem 3: 0, 2, 8, 8, 8 (5, total so far 2+5 = 7)
Stem 4: 0, 3, 4, 5, 6, 7, 8, 8 (8, total so far 7 + 8 = 15)
Stem 5: 0, 1, 2, 2 (4, total 15+4 = 19)
List all non - graduate data points in order:
21, 23, 30, 32, 38, 38, 38, 40, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 52
The 10th value is 44.
Since 40 < 44, graduates have a smaller median? Wait, no, wait the question is which group had the greater median. Wait, I think I made a mistake in graduate data. Wait, let's re - list graduate data:
Wait the stem - leaf for graduates:
Stem 2: 8,7,2 (so 22, 27, 28)
Stem 3: 5,3,2,2 (32, 33, 35? No, 2,2,3,5: 32, 32, 33, 35)
Stem 4: 6,5,4,3,1,0,0 (40, 40, 41, 43, 44, 45, 46)
Stem 5: 8,0 (50, 58)
So the data points in order: 22, 27, 28, 32, 32, 33, 35, 40, 40, 41, 43, 44, 45, 46, 50, 58
Number of data points: 3+4 + 7+2=16. The 8th term is 40 (index 8, starting from 1), the 9th term is 40. Median: 40.
Non - graduates: 19 data points. The median is the 10th term.
Data points:
Stem 2: 1,3 (21, 23)
Stem 3: 0,2,8,8,8 (30, 32, 38, 38, 38)
Stem 4: 0,3,4,5,6,7,8,8 (40, 43, 44, 45, 46, 47, 48, 48)
Stem 5: 0,1,2,2 (50, 51, 52, 52)
Order: 21,23,30,32,38,38,38,40,43,44,45,46,47,48,48,50,51,52,52
10th term is 44. So non - graduates have a greater median? Wait, no, the question is which group had the greater median. Wait, 40 (graduates) vs 44 (non - graduates). So non - graduates? Wait, no, maybe I messed up the stem - leaf reading. Wait the stem - leaf for graduates: the first stem is 2, with leaves 8,7,2? Wait, stem - leaf is stem | leaf, so for graduates, stem 2: leaves 2,7,8, so data is 22, 27, 28. Stem 3: leaves 2,2,3,5: 32, 32, 33, 35. Stem 4: leaves 0,0,1,3,4,5,6: 40, 40, 41, 43, 44, 45, 46. Stem 5: leaves 0,8: 50, 58. So 3+4 + 7+2 = 16. Correct.
Non - graduates: stem 2: leaves 1,3: 21,23. Stem 3: leaves 0,2,8,8,8: 30,32,38,38,38. Stem 4: leaves 0,3,4,5,6,7,8,8: 40,43,44,45,46,47,48,48. Stem 5: leaves 0,1,2,2: 50,51,52,52. 2 + 5+8 + 4=19. Correct.
So median for graduates: (40 + 40)/2 = 40. Median for non - graduates: 44. So non - graduates have greater median? Wait, but the options are Graduates, Non - graduates, sam…

Step1: Count responses in 20s for Graduates

For graduates, stem 2 (tens place 2) has leaves 2,7,8. So number of responses in 20s: 3.

Step2: Count responses in 20s for Non - graduates

For non - graduates, stem 2 has leaves 1,3. So number of responses in 20s: 2.
Since 3>2, graduates have more responses in the 20s.

Step1: Find range for Graduates

Range = maximum - minimum. For graduates, minimum value is 22 (from stem 2, leaf 2), maximum value is 58 (from stem 5, leaf 8). So range = $58 - 22=36$.

Step2: Find range for Non - graduates

For non - graduates, minimum value is 21 (from stem 2, leaf 1), maximum value is 52 (from stem 5, leaf 2). So range = $52 - 21 = 31$.

Answer:

Non - graduates

Part (b)