Question

temperatures
(in °f)
105 117 111 113 111
109 106 104 107 102
103 100 111 98 101
(a) complete the grouped frequency distribution for
the data. (note that the class width is 5.)
temperatures
(in °f)
frequency
98 to 102
103 to 107
108 to 112
113 to 117
(b) construct a histogram for the data.
frequency
6
5
4
3
2
1
0
98 to 102 103 to 107 108 to 112 113 to 117
temperatures (in °f)

Explanation:

Step1: Count 98 - 102

Check each data point: 100, 98, 101, 102, 104? No, 98,100,101,102, and 104? Wait, 98,100,101,102, and 104 is 103 - 107. Wait data: 105,117,111,113,111,109,106,104,107,102,103,100,111,98,101. Let's list all: 98,100,101,102,103,104,105,106,107,109,111,111,111,113,117. Now 98 - 102: 98,100,101,102. Wait 103? No, 98,100,101,102: that's 4? Wait 98,100,101,102, and 103? No, 98-102 is 98,100,101,102 (values ≤102). Wait 103 is 103-107. So 98,100,101,102: 4? Wait no, 98,100,101,102, and 103? No, 98-102 includes 98,100,101,102. Wait data points: 98,100,101,102,103,104,105,106,107,109,111,111,111,113,117. So 98-102: 98,100,101,102 → 4? Wait 103 is 103-107. 103,104,105,106,107: that's 5 (103,104,105,106,107). Then 108-112: 109,111,111,111 → 4 (109, three 111s). 113-117: 113,117 → 2? Wait no, 113,117, and 113? Wait data: 113,117. Wait 113 is 113-117, 117 is too. Wait 113,117: and 113? Wait data has 113 once, 117 once. Wait no, data: 113,117, and 111s, 109. Wait let's recount:

98 - 102: 98,100,101,102 → 4? Wait 103 is 103-107. 103,104,105,106,107 → 5 (103,104,105,106,107: that's 5 numbers). 108-112: 109,111,111,111 → 4 (109, three 111s: total 4). 113-117: 113,117 → 2? Wait no, 113,117, and 113? Wait data has 113 once, 117 once. Wait 113 is 113-117, 117 is 113-117. Wait 113,117: that's 2? Wait no, data: 113,117, and 111s, 109. Wait let's list each class:

- 98 to 102 (inclusive? Or exclusive? Class width 5, so 98 ≤ x < 103? Wait class width 5, so 98-102 (98 ≤ x < 103), 103-107 (103 ≤ x < 108), 108-112 (108 ≤ x < 113), 113-117 (113 ≤ x < 118).

So 98 ≤ x < 103: 98,100,101,102 (x=98,100,101,102: 4 values? Wait 103 is 103-107. Wait 103 is ≥103, so 98,100,101,102: 4? Wait 103 is 103, so 103 is in 103-107. Then 103,104,105,106,107: 103,104,105,106,107 → 5 (x=103,104,105,106,107: 5 values). 108 ≤ x < 113: 109,111,111,111 (x=109,111,111,111: 4 values, since 113 is ≥113). 113 ≤ x < 118: 113,117 (x=113,117: 2? Wait 113,117, and 113? Wait data has 113 once, 117 once. Wait 113 is 113, 117 is 117. So 2? Wait no, data: 113,117, and 111s, 109. Wait 113 is 113, 117 is 117, and 113? Wait original data: 105,117,111,113,111,109,106,104,107,102,103,100,111,98,101. Let's count each class:

98 to 102: numbers between 98 and 102 (inclusive? Or 98 ≤ x < 103). So 98,100,101,102. Wait 103 is 103-107. So 98,100,101,102: that's 4? Wait 100,101,102,98: 4 numbers.

103 to 107: 103,104,105,106,107. Let's check data: 103,104,105,106,107. Yes, 103 (from data: 103), 104,105,106,107. So 5 numbers.

108 to 112: 108 ≤ x < 113. So 109,111,111,111. Data: 109,111,111,111. That's 4 numbers (109, three 111s).

113 to 117: 113 ≤ x < 118. So 113,117. Data: 113,117. Wait 113 is once, 117 is once. Wait 113,117: and 113? Wait data has 113 once, 117 once. Wait 113 is 113, 117 is 117. So 2? Wait no, data has 113,117, and 113? Wait original data: 113 is present once, 117 once. So 113,117: 2? Wait no, 113 is 113, 117 is 117, and 113? Wait no, the data is: 105,117,111,113,111,109,106,104,107,102,103,100,111,98,101. So 113 is once, 117 is once. So 113,117: 2? Wait 113 is 113, 117 is 117, and 113? No, only one 113. So 113,117: 2? Wait 113 to 117: 113,117, and 113? No, data has 113 once, 117 once. So 2? Wait no, 113,117, and 113? Wait no, the data points are: 98,100,101,102,103,104,105,106,107,109,111,111,111,113,117. So 113,117: that's 2? Wait 113 is 113, 117 is 117. So 2. Wait but 113 is 113, 117 is 117, and 113? No, only one 113. So 2.

Wait let's recount:

98-102: 98,100,101,102 → 4

103-107: 103,104,105,106,107 → 5

108-112: 109,111,111,111 → 4 (109, three 111s)

113-117: 113,117 → 2? Wait 113,117, and 113? No, only one 113. Wait 113,117: 2. Wait but 113 is 113, 117 is 117, and 113? No, the data has 113 once, 117 once. So 2.

Wait but let's check again. Data points:

1. 105

2. 117

3. 111

4. 113

5. 111

6. 109

7. 106

8. 104

9. 107

10. 102

11. 103

12. 100

13. 111

14. 98

15. 101

Now categorize each:

- 98-102: points 10 (102), 12 (100), 14 (98), 15 (101). So 4 points (10,12,14,15).

- 103-107: points 11 (103), 8 (104), 1 (105), 7 (106), 9 (107). So 5 points (1,7,8,9,11).

- 108-112: points 6 (109), 3 (111), 5 (111), 13 (111). So 4 points (3,5,6,13).

- 113-117: points 4 (113), 2 (117). So 2 points (2,4).

So frequencies: 4,5,4,2.

Step2: Construct Histogram

For each class, the height is the frequency. So:

- 98-102: frequency 4 → bar height 4

- 103-107: frequency 5 → bar height 5

- 108-112: frequency 4 → bar height 4

- 113-117: frequency 2 → bar height 2

Answer:

(a) Frequencies: 98 to 102: 4; 103 to 107: 5; 108 to 112: 4; 113 to 117: 2.

(b) Histogram bars: 98-102 (height 4), 103-107 (height 5), 108-112 (height 4), 113-117 (height 2).