Question

the data set represents wait times (in minutes) for various services at a state’s department of motor vehicles locations. find the percentile that corresponds to a wait time of 18 minutes. a wait time of 18 minutes corresponds to about a percentile of \square. (round to the nearest whole number as needed.) \

$$\begin{tabular}{ccc} 3 & 21 & 6 \\\\ 20 & 1 & 6 \\\\ 10 & 3 & 2 \\\\ 13 & 3 & 4 \\\\ 14 & 5 & 14 \\\\ 11 & 5 & 15 \\\\ 14 & 6 & 16 \\\\ 17 & 10 & 4 \\\\ 4 & 1 & 19 \\\\ 27 & 22 & 3 \\\\ 4 & 23 & 19 \\\\ 8 & 10 & 26 \\\\ 4 & 8 & 5 \\\\ 3 & 7 & 3 \\\\ 26 & 2 & 4 \\\\ 18 & 1 & 7 \\\\ \\end{tabular}$$

Explanation:

Step1: Organize the data

First, we need to list out all the wait times and sort them in ascending order. Let's list the data: 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 10, 10, 11, 13, 14, 14, 14, 15, 16, 17, 18, 19, 19, 21, 22, 23, 26, 26, 27. Wait, let's count the number of data points. Let's check the original data:

First column (wait times? Wait, maybe the first column is the wait times? Wait, the data is presented as:

First column: 3, 20, 10, 13, 14, 11, 14, 17, 4, 27, 4, 8, 4, 3, 26, 18

Second column: 21, 1, 3, 3, 5, 5, 6, 10, 1, 22, 23, 10, 8, 7, 2, 1

Third column: 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 5, 3, 4, 7

Wait, maybe I misread. Let's count the number of data points. Let's list all the numbers:

From first column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18 (16 numbers)

Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1 (16 numbers)

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7 (16 numbers)

Total data points: 16+16+16=48.

Now, let's sort all the data in ascending order:

1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,6,7,7,8,8,10,10,10,11,13,14,14,14,14,15,16,17,18,19,19,20,21,22,23,26,26,27,27? Wait, no, let's list all 48 numbers:

Let's list each number:

From second column: 1,1,1 (wait, second column has three 1s? Wait second column: 21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1. So 1 appears at positions 2,9,16: three 1s.

First column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18. So 3: positions 1,14; 4: positions 9,11,13; 8: position 12; 10: position 3; 11: position 6; 13: position 4; 14: positions 5,7; 17: position 8; 18: position 16; 20: position 2; 26: position 15; 27: position 10.

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7. So 2: position 3; 3: positions 10,14; 4: positions 4,8,15; 5: position 13; 6: positions 1,2; 7: position 16; 14: position 5; 15: position 6; 16: position 7; 19: positions 9,11; 26: position 12.

Now let's list all numbers:

1 (from second col 2), 1 (second col 9), 1 (second col 16), 2 (third col 3), 2 (second col 15), 3 (first col 1), 3 (second col 3), 3 (second col 4), 3 (third col 10), 3 (third col 14), 4 (first col 9), 4 (first col 11), 4 (first col 13), 4 (third col 4), 4 (third col 8), 4 (third col 15), 5 (second col 5), 5 (second col 6), 5 (third col 13), 6 (third col 1), 6 (third col 2), 6 (second col 7), 7 (second col 14), 7 (third col 16), 8 (first col 12), 8 (second col 13), 10 (first col 3), 10 (second col 8), 10 (second col 12), 11 (first col 6), 13 (first col 4), 14 (first col 5), 14 (first col 7), 14 (third col 5), 14 (wait, first col 5 and 7: two 14s, third col 5: another 14? Wait first col 5:14, first col 7:14, third col 5:14. So three 14s? Wait no, first col 5:14, first col 7:14, third col 5:14. So three 14s. Then 15 (third col 6), 16 (third col 7), 17 (first col 8), 18 (first col 16), 19 (third col 9), 19 (third col 11), 20 (first col 2), 21 (second col 1), 22 (second col 10), 23 (second col 11), 26 (first col 15), 26 (third col 12), 27 (first col 10).

Wait, this is getting complicated. Maybe a better way: count the number of data points less than 18 and the number of data points equal to 18.

First, let's list all data points:

Let's list each number:

From first column (16 numbers): 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18

Second column (16 numbers):21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1

Third column (16 numbers):6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7

Now, let's list all 48 numbers:

1 (second col 2), 1 (second col 9), 1 (second col 16),

2 (third col 3), 2 (second col 15),

3 (first col 1), 3 (second col 3), 3 (second col 4), 3 (third col 10), 3 (third col 14),

4 (first col 9), 4 (first col 11), 4 (first col 13), 4 (third col 4), 4 (third col 8), 4 (third col 15),

5 (second col 5), 5 (second col 6), 5 (third col 13),

6 (third col 1), 6 (third col 2), 6 (second col 7),

7 (second col 14), 7 (third col 16),

8 (first col 12), 8 (second col 13),

10 (first col 3), 10 (second col 8), 10 (second col 12),

11 (first col 6),

13 (first col 4),

14 (first col 5), 14 (first col 7), 14 (third col 5),

15 (third col 6),

16 (third col 7),

17 (first col 8),

18 (first col 16),

19 (third col 9), 19 (third col 11),

20 (first col 2),

21 (second col 1),

22 (second col 10),

23 (second col 11),

26 (first col 15), 26 (third col 12),

27 (first col 10)

Now, let's count how many numbers are less than 18. Let's go through the list:

1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,10,10,10,11,13,14,14,14,15,16,17. Let's count these:

1 (3 times), 2 (2), 3 (5), 4 (6), 5 (3), 6 (3), 7 (2), 8 (2), 10 (3), 11 (1), 13 (1), 14 (3), 15 (1), 16 (1), 17 (1). Let's add: 3+2=5; 5+5=10; 10+6=16; 16+3=19; 19+3=22; 22+2=24; 24+2=26; 26+3=29; 29+1=30; 30+1=31; 31+3=34; 34+1=35; 35+1=36; 36+1=37. Wait, maybe a better way: count the number of elements before 18.

Looking at the list, the number 18 is at position (let's count the order):

Let's list the sorted data:

1,1,1,

2,2,

3,3,3,3,3,

4,4,4,4,4,4,

5,5,5,

6,6,6,

7,7,

8,8,

10,10,10,

11,

13,

14,14,14,

15,

16,

17,

18,

19,19,

20,

21,

22,

23,

26,26,

27

Now, let's count the number of elements less than 18:

From 1 to 17:

1: 3

2: 2 (total 5)

3: 5 (total 10)

4: 6 (total 16)

5: 3 (total 19)

6: 3 (total 22)

7: 2 (total 24)

8: 2 (total 26)

10: 3 (total 29)

11: 1 (total 30)

13: 1 (total 31)

14: 3 (total 34)

15: 1 (total 35)

16: 1 (total 36)

17: 1 (total 37)

So there are 37 numbers less than 18. Now, how many numbers are equal to 18? Looking at the data, the first column has 18, so only 1 number equal to 18.

The formula for percentile is:

Percentile = $\frac{\text{Number of values less than } x + 0.5 \times \text{Number of values equal to } x}{n} \times 100$

Where $n$ is the total number of data points.

Here, $x = 18$, number of values less than 18 = 37, number of values equal to 18 = 1, $n = 48$.

Step2: Calculate the percentile

Plug into the formula:

Percentile = $\frac{37 + 0.5 \times 1}{48} \times 100$

First, calculate the numerator: 37 + 0.5 = 37.5

Then, divide by 48: $\frac{37.5}{48} \approx 0.78125$

Multiply by 100: $0.78125 \times 100 = 78.125$

Round to the nearest whole number: 78.

Wait, but let's check the data again. Maybe I made a mistake in counting the number of data points less than 18. Let's re-examine the data.

Wait, the total number of data points: first column 16, second 16, third 16: 48. Let's list all data points with their values:

First column (wait times? Maybe the first column is the wait times? Wait, the problem says "the data set represents wait times (in minutes) for various services...". So maybe the first column is the wait times? Wait, the first column has numbers like 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18. Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1. Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7. So all three columns are wait times? So total 16*3=48 data points.

Now, let's list all 48 wait times:

First column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18

Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7

Now, let's sort all 48 numbers in ascending order:

1,1,1,

2,2,

3,3,3,3,3,

4,4,4,4,4,4,

5,5,5,

6,6,6,

7,7,

8,8,

10,10,10,

11,

13,

14,14,14,

15,

16,

17,

18,

19,19,

20,

21,

22,

23,

26,26,

27

Now, let's count the number of elements less than 18:

From 1 to 17:

1: 3 (positions 2,9,16 in second column)

2: 2 (third column 3, second column 15)

3: 5 (first column 1,14; second column 3,4; third column 10,14? Wait no, first column 1:3, first column 14:3; second column 3:3, second column 4:3; third column 10:3, third column 14:3. Wait that's 6? Wait first column 1:3, first column 14:3 (2); second column 3:3, second column 4:3 (2); third column 10:3, third column 14:3 (2). Total 6. Wait, earlier I thought 5, that was a mistake.

Let's recount:

1: 3 (second column: 1,1,1)

2: 2 (third column 3, second column 15)

3: 6 (first column 1,14; second column 3,4; third column 10,14)

4

Answer:

Step1: Organize the data

First, we need to list out all the wait times and sort them in ascending order. Let's list the data: 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 10, 10, 11, 13, 14, 14, 14, 15, 16, 17, 18, 19, 19, 21, 22, 23, 26, 26, 27. Wait, let's count the number of data points. Let's check the original data:

First column (wait times? Wait, maybe the first column is the wait times? Wait, the data is presented as:

First column: 3, 20, 10, 13, 14, 11, 14, 17, 4, 27, 4, 8, 4, 3, 26, 18

Second column: 21, 1, 3, 3, 5, 5, 6, 10, 1, 22, 23, 10, 8, 7, 2, 1

Third column: 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 5, 3, 4, 7

Wait, maybe I misread. Let's count the number of data points. Let's list all the numbers:

From first column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18 (16 numbers)

Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1 (16 numbers)

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7 (16 numbers)

Total data points: 16+16+16=48.

Now, let's sort all the data in ascending order:

1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,6,7,7,8,8,10,10,10,11,13,14,14,14,14,15,16,17,18,19,19,20,21,22,23,26,26,27,27? Wait, no, let's list all 48 numbers:

Let's list each number:

From second column: 1,1,1 (wait, second column has three 1s? Wait second column: 21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1. So 1 appears at positions 2,9,16: three 1s.

First column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18. So 3: positions 1,14; 4: positions 9,11,13; 8: position 12; 10: position 3; 11: position 6; 13: position 4; 14: positions 5,7; 17: position 8; 18: position 16; 20: position 2; 26: position 15; 27: position 10.

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7. So 2: position 3; 3: positions 10,14; 4: positions 4,8,15; 5: position 13; 6: positions 1,2; 7: position 16; 14: position 5; 15: position 6; 16: position 7; 19: positions 9,11; 26: position 12.

Now let's list all numbers:

1 (from second col 2), 1 (second col 9), 1 (second col 16), 2 (third col 3), 2 (second col 15), 3 (first col 1), 3 (second col 3), 3 (second col 4), 3 (third col 10), 3 (third col 14), 4 (first col 9), 4 (first col 11), 4 (first col 13), 4 (third col 4), 4 (third col 8), 4 (third col 15), 5 (second col 5), 5 (second col 6), 5 (third col 13), 6 (third col 1), 6 (third col 2), 6 (second col 7), 7 (second col 14), 7 (third col 16), 8 (first col 12), 8 (second col 13), 10 (first col 3), 10 (second col 8), 10 (second col 12), 11 (first col 6), 13 (first col 4), 14 (first col 5), 14 (first col 7), 14 (third col 5), 14 (wait, first col 5 and 7: two 14s, third col 5: another 14? Wait first col 5:14, first col 7:14, third col 5:14. So three 14s? Wait no, first col 5:14, first col 7:14, third col 5:14. So three 14s. Then 15 (third col 6), 16 (third col 7), 17 (first col 8), 18 (first col 16), 19 (third col 9), 19 (third col 11), 20 (first col 2), 21 (second col 1), 22 (second col 10), 23 (second col 11), 26 (first col 15), 26 (third col 12), 27 (first col 10).

Wait, this is getting complicated. Maybe a better way: count the number of data points less than 18 and the number of data points equal to 18.

First, let's list all data points:

Let's list each number:

From first column (16 numbers): 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18

Second column (16 numbers):21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1

Third column (16 numbers):6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7

Now, let's list all 48 numbers:

1 (second col 2), 1 (second col 9), 1 (second col 16),

2 (third col 3), 2 (second col 15),

3 (first col 1), 3 (second col 3), 3 (second col 4), 3 (third col 10), 3 (third col 14),

4 (first col 9), 4 (first col 11), 4 (first col 13), 4 (third col 4), 4 (third col 8), 4 (third col 15),

5 (second col 5), 5 (second col 6), 5 (third col 13),

6 (third col 1), 6 (third col 2), 6 (second col 7),

7 (second col 14), 7 (third col 16),

8 (first col 12), 8 (second col 13),

10 (first col 3), 10 (second col 8), 10 (second col 12),

11 (first col 6),

13 (first col 4),

14 (first col 5), 14 (first col 7), 14 (third col 5),

15 (third col 6),

16 (third col 7),

17 (first col 8),

18 (first col 16),

19 (third col 9), 19 (third col 11),

20 (first col 2),

21 (second col 1),

22 (second col 10),

23 (second col 11),

26 (first col 15), 26 (third col 12),

27 (first col 10)

Now, let's count how many numbers are less than 18. Let's go through the list:

1,1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,6,6,6,7,7,8,8,10,10,10,11,13,14,14,14,15,16,17. Let's count these:

1 (3 times), 2 (2), 3 (5), 4 (6), 5 (3), 6 (3), 7 (2), 8 (2), 10 (3), 11 (1), 13 (1), 14 (3), 15 (1), 16 (1), 17 (1). Let's add: 3+2=5; 5+5=10; 10+6=16; 16+3=19; 19+3=22; 22+2=24; 24+2=26; 26+3=29; 29+1=30; 30+1=31; 31+3=34; 34+1=35; 35+1=36; 36+1=37. Wait, maybe a better way: count the number of elements before 18.

Looking at the list, the number 18 is at position (let's count the order):

Let's list the sorted data:

1,1,1,

2,2,

3,3,3,3,3,

4,4,4,4,4,4,

5,5,5,

6,6,6,

7,7,

8,8,

10,10,10,

11,

13,

14,14,14,

15,

16,

17,

18,

19,19,

20,

21,

22,

23,

26,26,

27

Now, let's count the number of elements less than 18:

From 1 to 17:

1: 3

2: 2 (total 5)

3: 5 (total 10)

4: 6 (total 16)

5: 3 (total 19)

6: 3 (total 22)

7: 2 (total 24)

8: 2 (total 26)

10: 3 (total 29)

11: 1 (total 30)

13: 1 (total 31)

14: 3 (total 34)

15: 1 (total 35)

16: 1 (total 36)

17: 1 (total 37)

So there are 37 numbers less than 18. Now, how many numbers are equal to 18? Looking at the data, the first column has 18, so only 1 number equal to 18.

The formula for percentile is:

Percentile = $\frac{\text{Number of values less than } x + 0.5 \times \text{Number of values equal to } x}{n} \times 100$

Where $n$ is the total number of data points.

Here, $x = 18$, number of values less than 18 = 37, number of values equal to 18 = 1, $n = 48$.

Step2: Calculate the percentile

Plug into the formula:

Percentile = $\frac{37 + 0.5 \times 1}{48} \times 100$

First, calculate the numerator: 37 + 0.5 = 37.5

Then, divide by 48: $\frac{37.5}{48} \approx 0.78125$

Multiply by 100: $0.78125 \times 100 = 78.125$

Round to the nearest whole number: 78.

Wait, but let's check the data again. Maybe I made a mistake in counting the number of data points less than 18. Let's re-examine the data.

Wait, the total number of data points: first column 16, second 16, third 16: 48. Let's list all data points with their values:

First column (wait times? Maybe the first column is the wait times? Wait, the problem says "the data set represents wait times (in minutes) for various services...". So maybe the first column is the wait times? Wait, the first column has numbers like 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18. Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1. Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7. So all three columns are wait times? So total 16*3=48 data points.

Now, let's list all 48 wait times:

First column: 3,20,10,13,14,11,14,17,4,27,4,8,4,3,26,18

Second column:21,1,3,3,5,5,6,10,1,22,23,10,8,7,2,1

Third column:6,6,2,4,14,15,16,4,19,3,19,26,5,3,4,7

Now, let's sort all 48 numbers in ascending order:

1,1,1,

2,2,

3,3,3,3,3,

4,4,4,4,4,4,

5,5,5,

6,6,6,

7,7,

8,8,

10,10,10,

11,

13,

14,14,14,

15,

16,

17,

18,

19,19,

20,

21,

22,

23,

26,26,

27

Now, let's count the number of elements less than 18:

From 1 to 17:

1: 3 (positions 2,9,16 in second column)

2: 2 (third column 3, second column 15)

3: 5 (first column 1,14; second column 3,4; third column 10,14? Wait no, first column 1:3, first column 14:3; second column 3:3, second column 4:3; third column 10:3, third column 14:3. Wait that's 6? Wait first column 1:3, first column 14:3 (2); second column 3:3, second column 4:3 (2); third column 10:3, third column 14:3 (2). Total 6. Wait, earlier I thought 5, that was a mistake.

Let's recount:

1: 3 (second column: 1,1,1)

2: 2 (third column 3, second column 15)

3: 6 (first column 1,14; second column 3,4; third column 10,14)

4