Question

(in minutes)
26 29 18 16 33 19 18 34
35 27 31 23 31 24 30 28
(a) complete the grouped frequency distribution for
the data. (note that the class width is 4.)
shopping times
(in minutes) frequency
16 to 19
20 to 23
24 to 27
28 to 31
32 to 35
(b) construct a histogram for the data.
frequency
6
5
4
3
2
1
0
16 to 19 20 to 23 24 to 27 28 to 31 32 to 35
shopping times (in minutes)

Explanation:

Step1: Count 16 - 19

Data points: 16, 18, 18, 19. So frequency = 4.

Step2: Count 20 - 23

Data points: 23 (only 23? Wait, 20 - 23: check numbers. Given data: 26,29,18,16,33,19,18,34,35,27,31,23,31,24,30,28. Wait 20 - 23: numbers between 20 and 23 inclusive. Only 23? Wait no, 23 is 23, but 20 - 23: 20,21,22,23. So only 23? Wait no, maybe I missed. Wait the data: 23 is one. Wait maybe I made a mistake. Wait let's list all data: 16,18,18,19,23,24,26,27,28,30,31,31,33,34,35,29? Wait no, original data: first row: 26,29,18,16,33,19,18,34. Second row:35,27,31,23,31,24,30,28. So all data points: 26,29,18,16,33,19,18,34,35,27,31,23,31,24,30,28. Now categorize:

16 - 19: 16,18,18,19 → 4 values.

20 - 23: 23 (since 20,21,22,23: only 23) → 1? Wait no, 23 is 23, but maybe the class is 16 - 19 (width 4: 16 - 19, 20 - 23, 24 - 27, 28 - 31, 32 - 35). Wait 20 - 23: 20 to 23, so 20,21,22,23. So data points: 23 (only one). Wait 24 - 27: 24,26,27? Wait 24,26,27? Wait 24 is 24, 26,27. Wait 24 - 27: 24,25,26,27. So 24,26,27 → 3? Wait 24,26,27: 24 is 24, 26,27. Wait 24,26,27: three? Wait 24,26,27: 24,26,27. Wait 28 - 31: 28,30,31,31. Wait 28,30,31,31: 28,30,31,31 → 4? Wait 32 - 35: 33,34,35 → 3? Wait let's recheck:

16 - 19: 16,18,18,19 → 4.

20 - 23: 23 → 1? Wait no, maybe I messed up the class width. Wait class width 4: 16 - 19 (16 ≤ x < 20), 20 - 23 (20 ≤ x < 24), 24 - 27 (24 ≤ x < 28), 28 - 31 (28 ≤ x < 32), 32 - 35 (32 ≤ x < 36). Ah! That's the correct way: exclusive class boundaries. So 16 - 19: 16 ≤ x < 20. So data: 16,18,18,19 (all <20) → 4.

20 - 23: 20 ≤ x < 24. So data: 23 (23 <24) → 1? Wait no, 23 is 23, but 20 - 23 (exclusive upper: <24). So 23 is included. Any others? 24 is in 24 - 27. So 20 - 23: only 23? Wait no, maybe I missed. Wait the data: 23 is one. Wait 24 - 27: 24,26,27 (24 ≤ x <28). 24,26,27: three? Wait 24,26,27: 24,26,27. 28 - 31: 28,30,31,31 (28 ≤ x <32). 28,30,31,31: 4. 32 - 35: 33,34,35 (32 ≤ x <36). 33,34,35: 3. Wait but let's count all data points: 4 (16 - 19) +1 (20 - 23) +3 (24 - 27) +4 (28 - 31) +3 (32 - 35) = 15? Wait original data has 16 points (8 + 8). Oh, I missed one. Wait first row: 8, second row:8. Total 16. So my count is wrong. Let's list all 16:

First row: 26,29,18,16,33,19,18,34 (8)

Second row:35,27,31,23,31,24,30,28 (8)

Now:

16 - 19 (16 ≤ x <20): 16,18,18,19 → 4 (correct, 4 points)

20 - 23 (20 ≤ x <24): 23 (only 23? Wait 20,21,22,23: but data has 23. Wait no, maybe 20 - 23 includes 20,21,22,23. But data has no 20,21,22. Only 23. So 1 point? But 4 +1 +...=16? Let's check 24 - 27 (24 ≤ x <28): 24,26,27,26? Wait no, 24,26,27: 24,26,27. Wait 24 is 24, 26,27. Wait 24,26,27: three? Wait 28 - 31 (28 ≤ x <32): 28,30,31,31 → 4. 32 - 35 (32 ≤ x <36): 33,34,35 → 3. Then 26 and 29: wait 26 is in 24 - 27 (24 ≤x <28: 26 is 26 <28, yes. 29: 29 is in 28 - 31? No, 28 - 31 is <32, 29 is 29 <32, so 29 is in 28 - 31? Wait 28 - 31: 28 ≤x <32. So 28,29,30,31. Oh! I made a mistake here. So 28 - 31 includes 28,29,30,31. So data points: 28,29,30,31,31. Wait 28,29,30,31,31: that's 5? Wait no, let's re-express each data point:

1. 26: 24 - 27 (24 ≤x <28)
2. 29: 28 - 31 (28 ≤x <32)
3. 18: 16 - 19
4. 16: 16 - 19
5. 33: 32 - 35
6. 19: 16 - 19
7. 18: 16 - 19
8. 34: 32 - 35
9. 35: 32 - 35
10. 27: 24 - 27
11. 31: 28 - 31
12. 23: 20 - 23
13. 31: 28 - 31
14. 24: 24 - 27
15. 30: 28 - 31
16. 28: 28 - 31

Now let's count each class:

16 - 19: points 3,4,6,7 → 18,16,19,18 → 4 (correct)

20 - 23: point 12 → 23 → 1 (correct)

24 - 27: points 1,10,14 → 26,27,24 → 3 (correct)

28 - 31: points 2,11,13,15,16 → 29,31,31,30,28 → 5? Wait no, 29,31,31,30,28: that's 5. But 4 (16-19) +1 (20-23) +3 (24-27) +5 (28-31) +3 (32-35) = 16. Ah, I missed 29 earlier. So 28 - 31 has 5? Wait no, 28,29,30,31,31: five points.

32 - 35: points 5,8,9 → 33,34,35 → 3 (correct)

Wait but the original problem's frequency table has 5 classes: 16-19,20-23,24-27,28-31,32-35.

So:

16 - 19: 4

20 - 23: 1

24 - 27: 3 (24,26,27)

28 - 31: 5 (28,29,30,31,31)

32 - 35: 3 (33,34,35)

Wait but let's check again:

Data points:

1. 26 → 24-27

2. 29 → 28-31

3. 18 → 16-19

4. 16 → 16-19

5. 33 → 32-35

6. 19 → 16-19

7. 18 → 16-19

8. 34 → 32-35

9. 35 → 32-35

10. 27 → 24-27

11. 31 → 28-31

12. 23 → 20-23

13. 31 → 28-31

14. 24 → 24-27

15. 30 → 28-31

16. 28 → 28-31

Now count each class:

16-19: points 3,4,6,7 → 4 ✔️

20-23: point 12 → 1 ✔️

24-27: points 1,10,14 → 3 ✔️

28-31: points 2,11,13,15,16 → 5 ✔️

32-35: points 5,8,9 → 3 ✔️

Now, for part (b), the histogram: x-axis is shopping times (16-19,20-23,24-27,28-31,32-35), y-axis frequency. So the bars should have heights: 4,1,3,5,3.

But since the problem is to complete the frequency distribution (part a) and construct histogram (part b), but the user probably needs the frequency counts.

Step3: Final Frequencies

16-19: 4

20-23: 1

24-27: 3

28-31: 5

32-35: 3

Answer:

For part (a), the frequencies are:

16 to 19: 4

20 to 23: 1

24 to 27: 3

28 to 31: 5

32 to 35: 3

(For the histogram, draw bars with these heights for each class interval.)