Question

the data set represents wait times (in minutes) for various services at a state’s department of motor vehicles locations. which wait times are between the 25th and 75th percentiles? 18, 21, 1, 3, 5, 5, 6, 10, 1, 22, 23, 10, 6, 7, 1, 6, 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 3, 7, 6, 10, 9, 10, 20, 18, 20, 10, 13, 14, 11, 14, 17, 4, 27, 4, 8, 4, 26. wait times between □ minutes and □ minutes are between the 25th and 75th percentiles.

Explanation:

Step1: Count the number of data points

First, we count the total number of wait times. Let's list out all the data points: 18, 21, 1, 3, 5, 5, 6, 10, 1, 22, 23, 10, 6, 7, 1, 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 3, 7, 6, 10, 9, 10, 20, 18, 20, 10, 13, 14, 11, 14, 17, 4, 27, 4, 8, 4, 26. Let's count them: let's do it step by step. First row: 18,21,1,3,5,5,6,10,1,22,23,10,6,7,1,6,6 (17); second row: 2,4,14,15,16,4,19,3,19,26,3,7,6,10,9 (15); third row: 10,20,18,20,10,13,14,11,14,17,4,27,4,8,4,26 (16). Wait, maybe a better way: let's count the number of elements. Let's list all:

18, 21, 1, 3, 5, 5, 6, 10, 1, 22, 23, 10, 6, 7, 1, 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 3, 7, 6, 10, 9, 10, 20, 18, 20, 10, 13, 14, 11, 14, 17, 4, 27, 4, 8, 4, 26. Let's count: 1.18, 2.21, 3.1, 4.3, 5.5, 6.5, 7.6, 8.10, 9.1, 10.22, 11.23, 12.10, 13.6, 14.7, 15.1, 16.6, 17.6, 18.2, 19.4, 20.14, 21.15, 22.16, 23.4, 24.19, 25.3, 26.19, 27.26, 28.3, 29.7, 30.6, 31.10, 32.9, 33.10, 34.20, 35.18, 36.20, 37.10, 38.13, 39.14, 40.11, 41.14, 42.17, 43.4, 44.27, 45.4, 46.8, 47.4, 48.26. So total \( n = 48 \) data points.

Step2: Find the position of the 25th and 75th percentiles

The formula for the position of the \( p \)-th percentile is \( i = \frac{p}{100} \times n \).

For the 25th percentile (\( p = 25 \)):
\( i_{25} = \frac{25}{100} \times 48 = 0.25 \times 48 = 12 \)

For the 75th percentile (\( p = 75 \)):
\( i_{75} = \frac{75}{100} \times 48 = 0.75 \times 48 = 36 \)

Step3: Sort the data set

Now, we need to sort the data in ascending order. Let's list the sorted data:

First, list all unique values and count their frequencies:

1: appears 3 times (positions 1,9,15)
2: 1 time (18)
3: 3 times (4,25,28)
4: 5 times (19,23,43,45,47)
5: 2 times (5,6)
6: 5 times (7,13,16,17,30)
7: 2 times (14,29)
8: 1 time (46)
9: 1 time (32)
10: 5 times (8,12,31,33,37)
11: 1 time (40)
13: 1 time (38)
14: 3 times (20,39,41)
15: 1 time (21)
16: 1 time (22)
17: 1 time (42)
18: 2 times (1,35)
19: 2 times (24,26)
20: 2 times (34,36)
21: 1 time (2)
22: 1 time (10)
23: 1 time (11)
26: 2 times (27,48)
27: 1 time (44)

Now, let's sort them in ascending order:

1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 8, 9, 10, 10, 10, 10, 10, 11, 13, 14, 14, 14, 15, 16, 17, 18, 18, 19, 19, 20, 20, 21, 22, 23, 26, 26, 27

Wait, let's check the count. Let's count the sorted data:

1 (3), 2 (1), 3 (3), 4 (5), 5 (2), 6 (5), 7 (2), 8 (1), 9 (1), 10 (5), 11 (1), 13 (1), 14 (3), 15 (1), 16 (1), 17 (1), 18 (2), 19 (2), 20 (2), 21 (1), 22 (1), 23 (1), 26 (2), 27 (1). Let's sum the frequencies: 3+1=4; +3=7; +5=12; +2=14; +5=19; +2=21; +1=22; +1=23; +5=28; +1=29; +1=30; +3=33; +1=34; +1=35; +1=36; +2=38; +2=40; +2=42; +1=43; +1=44; +1=45; +2=47; +1=48. Perfect, 48 data points.

Now, the sorted list (indexes from 1 to 48):

1: 1

2: 1

3: 1

4: 2

5: 3

6: 3

7: 3

8: 4

9: 4

10: 4

11: 4

12: 4 (25th percentile position \( i_{25}=12 \), so the 12th element)

13: 5

14: 5

15: 6

16: 6

17: 6

18: 6

19: 6

20: 7

21: 7

22: 8

23: 9

24: 10

25: 10

26: 10

27: 10

28: 10

29: 11

30: 13

31: 14

32: 14

33: 14

34: 15

35: 16

36: 17 (75th percentile position \( i_{75}=36 \), so the 36th element)

37: 18

38: 18

39: 19

40: 19

41: 20

42: 20

43: 21

44: 22

45: 23

46: 26

47: 26

48: 27

Wait, let's list the sorted data with their indices (1-based):

1: 1

2: 1

3: 1

4: 2

5: 3

6: 3

7: 3

8: 4

9: 4

10: 4

11: 4

12: 4 (element at index 12: 4? Wait no, wait the frequency of 4 is 5, so the first 4 is at index 8, then 9,10,11,12. Wait, let's list the sorted data properly:

Indices 1-3: 1,1,1

Index 4: 2

Indices 5-7: 3,3,3

Indices 8-12: 4,4,4,4,4 (since 5 fours: 8,9,10,11,12)

Indices 13-14: 5,5

Indices 15-19: 6,6,6,6,6 (5 sixes: 15,16,17,18,19)

Indices 20-21: 7,7

Index 22: 8

Index 23: 9

Indices 24-28: 10,10,10,10,10 (5 tens: 24,25,26,27,28)

Index 29: 11

Index 30: 13

Indices 31-33: 14,14,14 (3 fourteens: 31,32,33)

Index 34: 15

Index 35: 16

Index 36: 17 (element at index 36: 17)

Indices 37-38: 18,18 (2 eighteens: 37,38)

Indices 39-40: 19,19 (2 nineteens: 39,40)

Indices 41-42: 20,20 (2 twenties: 41,42)

Index 43: 21

Index 44: 22

Index 45: 23

Indices 46-47: 26,26 (2 twentysixes: 46,47)

Index 48: 27

Ah, I see, I made a mistake earlier. Let's correct the sorted list:

1: 1

2: 1

3: 1

4: 2

5: 3

6: 3

7: 3

8: 4

9: 4

10: 4

11: 4

12: 4 (25th percentile: index 12, value 4)

13: 5

14: 5

15: 6

16: 6

17: 6

18: 6

19: 6

20: 7

21: 7

22: 8

23: 9

24: 10

25: 10

26: 10

27: 10

28: 10

29: 11

30: 13

31: 14

32: 14

33: 14

34: 15

35: 16

36: 17 (75th percentile: index 36, value 17)

37: 18

38: 18

39: 19

40: 19

41: 20

42: 20

43: 21

44: 22

45: 23

46: 26

47: 26

48: 27

Wait, but let's verify the count. From index 1-3: 3 elements (1s)

Index 4: 1 element (2) → total 4

Indices 5-7: 3 elements (3s) → total 7

Indices 8-12: 5 elements (4s) → total 12 (7+5=12)

Indices 13-14: 2 elements (5s) → total 14 (12+2=14)

Indices 15-19: 5 elements (6s) → total 19 (14+5=19)

Indices 20-21: 2 elements (7s) → total 21 (19+2=21)

Index 22: 1 element (8) → total 22 (21+1=22)

Index 23: 1 element (9) → total 23 (22+1=23)

Indices 24-28: 5 elements (10s) → total 28 (23+5=28)

Index 29: 1 element (11) → total 29 (28+1=29)

Index 30: 1 element (13) → total 30 (29+1=30)

Indices 31-33: 3 elements (14s) → total 33 (30+3=33)

Index 34: 1 element (15) → total 34 (33+1=34)

Index 35: 1 element (16) → total 35 (34+1=35)

Index 36: 1 element (17) → total 36 (35+1=36)

Indices 37-38: 2 elements (18s) → total 38 (36+2=38)

Indices 39-40: 2 elements (19s) → total 40 (38+2=40)

Indices 41-42: 2 elements (20s) → total 42 (40+2=42)

Index 43: 1 element (21) → total 43 (42+1=43)

Index 44: 1 element (22) → total 44 (43+1=44)

Index 45: 1 element (23) → total 45 (44+1=45)

Indices 46-47: 2 elements (26s) → total 47 (45+2=47)

Index 48: 1 element (27) → total 48 (47+1=48). Perfect, that's correct.

Step4: Identify the values at the 25th and 75th percentiles

The 25th percentile is at index 12, which is 4.

The 75th percentile is at index 36, which is 17.

So the wait times between the 25th and 75th percentiles are between 4 minutes and 17 minutes.

Answer:

Wait times between \(\boldsymbol{4}\) minutes and \(\boldsymbol{17}\) minutes are between the 25th and 75th percentiles.