Question

the stem-and-leaf plot below gives the typing speeds of two groups of students. there were 19 students in group 1 and 16 in group 2. use the plot to answer the questions.
typing speed (words per minute)
group 1 group 2
8 4 4 1 | 3 | 5 7
9 5 4 3 3 3 0 0 | 4 | 0 5 6 7 8 9
9 8 6 5 5 | 5 | 3 3 7 9 9
5 1 | 6 | 0 1 1
(a) which group had the greater median typing speed?
○ group 1 ○ group 2 ○ the medians were the same
(b) what were the ranges of typing speeds for the two groups?
group 1
words per minute
group 2
words per minute
(c) which group had more typing speeds in the 60s?
○ group 1 ○ group 2 ○ each had the same

Explanation:

Part (a)

Step 1: Find median of Group 1

Group 1 has 19 students. The median is the 10th value (since \((19 + 1)/2 = 10\)). Let's list the data from the stem - leaf plot:

• Stem 3: 1, 4, 4, 8 (4 values)
• Stem 4: 0, 0, 3, 3, 3, 4, 5, 9 (8 values, cumulative: \(4 + 8=12\) values, so the 10th value is in stem 4? Wait, no. Wait, stem 3 has 4 values, stem 4 has 8 values (total 12), so the 10th value is in stem 4. Wait, let's count properly:

Values in Group 1:
Stem 3: 1, 4, 4, 8 (4)
Stem 4: 0, 0, 3, 3, 3, 4, 5, 9 (8, total \(4 + 8 = 12\))
Stem 5: 5, 5, 6, 8, 9 (5, total \(12+5 = 17\))
Stem 6: 1, 5 (2, total \(17 + 2=19\))
Wait, we need the 10th value. Let's count: 1 (1), 4 (2), 4 (3), 8 (4), 0 (5), 0 (6), 3 (7), 3 (8), 3 (9), 4 (10). So the 10th value is 4 (from stem 4). Wait, no, the stem - leaf plot for Group 1:
Stem 3: 1, 4, 4, 8 (4 values)
Stem 4: 0, 0, 3, 3, 3, 4, 5, 9 (8 values, positions 5 - 12)
So the 10th value is the 6th value in stem 4? Wait, no. The first 4 values are stem 3. Then the next 8 are stem 4. So the 10th value is the \(10 - 4=6\)th value in stem 4. The values in stem 4 (sorted) are 0, 0, 3, 3, 3, 4, 5, 9. The 6th value is 4. Wait, no, 0 (1), 0 (2), 3 (3), 3 (4), 3 (5), 4 (6). Yes, so median of Group 1 is 4? Wait, no, wait the stem - leaf plot for Group 1:
Wait, the stem - leaf plot is written as:
Group 1:
3 | 8 4 4 1 (wait, maybe I read the stem - leaf plot wrong. Maybe the stem is on the left, and the leaves are on the right, but maybe the order is reversed? Wait, stem - leaf plot: the stem is the tens place, and the leaves are the units place. And the leaves are usually written in ascending order. So maybe the Group 1 stem - leaf is:
Stem 3: 1, 4, 4, 8 (so values 31, 34, 34, 38)
Stem 4: 0, 0, 3, 3, 3, 4, 5, 9 (values 40, 40, 43, 43, 43, 44, 45, 49)
Stem 5: 5, 5, 6, 8, 9 (values 55, 55, 56, 58, 59)
Stem 6: 1, 5 (values 61, 65)
Now, let's list all 19 values:
31, 34, 34, 38,
40, 40, 43, 43, 43, 44, 45, 49,
55, 55, 56, 58, 59,
61, 65
Now, the 10th value (since \(n = 19\), median is the \((19 + 1)/2=10\)th term). Let's count:
1: 31, 2: 34, 3: 34, 4: 38,
5: 40, 6: 40, 7: 43, 8: 43, 9: 43, 10: 44. So median of Group 1 is 44.

Step 2: Find median of Group 2

Group 2 has 16 students. The median is the average of the 8th and 9th values (since \((16)/2 = 8\), so median is \(\frac{\text{8th value}+\text{9th value}}{2}\)).
Values in Group 2:
Stem 3: 5, 7 (values 35, 37)
Stem 4: 0, 5, 6, 7, 8, 9 (values 40, 45, 46, 47, 48, 49)
Stem 5: 3, 3, 7, 9, 9 (values 53, 53, 57, 59, 59)
Stem 6: 0, 1, 1 (values 60, 61, 61)
List all 16 values:
35, 37,
40, 45, 46, 47, 48, 49,
53, 53, 57, 59, 59,
60, 61, 61
Now, the 8th value: Let's count:
1: 35, 2: 37,
3: 40, 4: 45, 5: 46, 6: 47, 7: 48, 8: 49
The 9th value: 53
Median of Group 2 is \(\frac{49 + 53}{2}=\frac{102}{2}=51\)? Wait, no, wait:
Wait, stem 3: 2 values, stem 4: 6 values (total \(2 + 6 = 8\) values), stem 5: 5 values (total \(8+5 = 13\)), stem 6: 3 values (total \(13 + 3=16\)). So the 8th value is the last value in stem 4 (49), and the 9th value is the first value in stem 5 (53). So median of Group 2 is \(\frac{49+53}{2}=51\)? Wait, no, that can't be. Wait, I think I made a mistake in reading the stem - leaf plot. Wait, Group 2's stem - leaf:
Stem 3: 5,7 (35, 37)
Stem 4: 0,5,6,7,8,9 (40, 45, 46, 47, 48, 49)
Stem 5: 3,3,7,9,9 (53, 53, 57, 59, 59)
Stem 6: 0,1,1 (60, 61, 61)
Now, let's list the values in order:
35, 37,
40, 45, 46, 47, 48, 49,
53, 53, 57, 59, 59,
60, 61, 61
Now, the 8th value: Let's count: 1 (35), 2 (37), 3 (40), 4 (45), 5 (46), 6 (47), 7 (48), 8 (49)
The 9th value: 53
So median…

Step 1: Range of Group 1

Range is maximum - minimum.
Minimum value in Group 1: 31 (from stem 3, leaf 1)
Maximum value in Group 1: 65 (from stem 6, leaf 5)
Range of Group 1: \(65 - 31=34\)

Step 2: Range of Group 2

Minimum value in Group 2: 35 (from stem 3, leaf 5)
Maximum value in Group 2: 61 (from stem 6, leaf 1)
Range of Group 2: \(61 - 35 = 26\)

Part (c)

Step 1: Count typing speeds in 60s for Group 1

Typing speeds in 60s: stem 6. Group 1 has leaves 1 and 5, so two values: 61 and 65.

Step 2: Count typing speeds in 60s for Group 2

Typing speeds in 60s: stem 6. Group 2 has leaves 0,1,1, so three values: 60, 61, 61.

So Group 2 has more typing speeds in the 60s.

(a) Answer: Group 2

(b) Answer: Group 1: 34, Group 2: 26

(c) Answer: Group 2

Answer:

Step 1: Count typing speeds in 60s for Group 1

Typing speeds in 60s: stem 6. Group 1 has leaves 1 and 5, so two values: 61 and 65.

Step 2: Count typing speeds in 60s for Group 2

Typing speeds in 60s: stem 6. Group 2 has leaves 0,1,1, so three values: 60, 61, 61.

So Group 2 has more typing speeds in the 60s.

(a) Answer: Group 2

(b) Answer: Group 1: 34, Group 2: 26

(c) Answer: Group 2