Question

the dot plot of time (in minutes) it takes employees to drive to work is shown below. find the mean, median, and mode of the data. if any measure cannot be found or does not represent the center of the data, explain why. find the mean. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the mean is
(type an integer or decimal rounded to one decimal place as needed.)
b. there is no mean for this data set

Explanation:

Step1: Count the number of dots for each value

- At 5: 1 dot
- At 7: 1 dot (assuming the second dot is at 7, since the first three are at 5,7,9? Wait, maybe the x-axis is labeled with 5, then spaces, then 10, 15, 20, 25, 30, 35. Let's re-express the dot plot:
Looking at the dot plot:
- 5: 1 dot
- 7: 1 dot (next after 5)
- 9: 1 dot (next after 7)
- 15: 3 dots (wait, the dots at 15: let's count properly. Let's list each value and its frequency:
Let's assume the x-axis has marks at 5, 6, 7, 8, 9, 10, ..., 15, 16, ..., 20, ..., 25, ..., 30, ..., 35.
From the plot:
- 5: 1
- 7: 1 (second dot)
- 9: 1 (third dot)
- 15: 3 (three dots)
- 16: 2 (two dots? Wait, the original plot: "dots at 5, then 7, then 9, then a gap, then at 15: three dots? Wait, maybe the x-axis is 5, 10, 15, 20, 25, 30, 35 with intervals. Wait, the first three dots are at 5, 7, 9? No, maybe the x-axis is labeled with 5, then each vertical line is 1 unit. So:

Let's list all values with their frequencies:

- 5: 1
- 7: 1 (second dot)
- 9: 1 (third dot)
- 15: 3 (three dots)
- 16: 2 (two dots? Wait, the original plot: "dots at 5, then 7, then 9, then a gap, then at 15: three dots (15,16,17? No, the dot plot has vertical lines. Let's re-express:

Wait, the correct way: let's count each dot's position:

First three dots: at 5, 7, 9 (each 1 unit apart? No, maybe 5, 10, 15? No, the first three are at 5, then next at 7, then 9, then a gap until 15. Then at 15: 3 dots (15,16,17? No, the dot plot shows:

Looking at the plot:

- 5: 1 dot
- 7: 1 dot (second)
- 9: 1 dot (third)
- 15: 3 dots (15,16,17? Wait, no, the vertical lines are for each minute. So:

Let's list all values:

5: 1

7: 1

9: 1

15: 3 (15,16,17? No, the dots at 15: three dots, so 15 appears 3 times? Wait, no, the dot plot: each dot is a data point. So:

Wait, maybe the x-axis is 5, 10, 15, 20, 25, 30, 35 with each mark representing a minute, and the dots are:

- 5: 1

- 7: 1 (next line)

- 9: 1 (next line)

- 15: 3 (three dots at 15,16,17? No, the plot shows:

After 9, there's a gap until 15, where there are three dots (15,16,17? No, the vertical lines are for each minute, so:

Let's count the number of dots at each x-value:

- x=5: 1

- x=7: 1

- x=9: 1

- x=15: 3 (three dots)

- x=16: 2 (two dots? Wait, the original plot: "dots at 5, then 7, then 9, then a gap, then at 15: three dots (15,16,17? No, the user's plot: "dots at 5, then 7, then 9, then a gap, then at 15: three dots (15,16,17? Wait, no, the correct way is to look at the positions:

Wait, maybe the x-axis is labeled with 5, 10, 15, 20, 25, 30, 35, and the dots are:

- 5: 1

- 7: 1 (between 5 and 10)

- 9: 1 (between 5 and 10)

- 15: 3 (at 15)

- 16: 2 (at 16)

- 19: 1 (at 19)

- 20: 2 (at 20)

- 21: 1 (at 21)

- 25: 1 (at 25)

- 28: 1 (at 28)

- 34: 1 (at 34)

- 37: 1 (at 37? No, the last dot is at 35? Wait, the x-axis ends at 35. So:

Wait, let's re-express the dot plot with correct frequencies:

- 5: 1

- 7: 1

- 9: 1

- 15: 3 (three dots)

- 16: 2 (two dots)

- 19: 1 (one dot)

- 20: 2 (two dots)

- 21: 1 (one dot)

- 25: 1 (one dot)

- 28: 1 (one dot)

- 34: 1 (one dot)

- 37: No, the last dot is at 35? Wait, the x-axis is 5, 10, 15, 20, 25, 30, 35. So maybe the dots are:

- 5: 1

- 7: 1

- 9: 1

- 15: 3

- 16: 2

- 19: 1

- 20: 2

- 21: 1

- 25: 1

- 28: 1

- 34: 1

- 37: No, 35 is the last mark. So 35: 1.

Wait, maybe I made a mistake. Let's count the total number of dots:

1 (5) + 1 (7) + 1 (9) + 3 (15) + 2 (16) + 1 (19) + 2 (20) + 1 (21) + 1 (25) + 1 (28) + 1 (34) + 1 (35) = Let's calculate:

1+1=2; +1=3; +3=6; +2=8; +1=9; +2=11; +1=12; +1=13; +1=14; +1=15; +1=16. So 16 data points.

Now, to find the mean, we need to calculate the sum of all data points divided by 16.

Let's list each data point:

5, 7, 9, 15,15,15, 16,16, 19, 20,20, 21, 25, 28, 34, 35.

Wait, let's check the frequencies:

- 5: 1 → 5

- 7: 1 → 7

- 9: 1 → 9

- 15: 3 → 15,15,15

- 16: 2 → 16,16

- 19: 1 → 19

- 20: 2 → 20,20

- 21: 1 → 21

- 25: 1 → 25

- 28: 1 → 28

- 34: 1 → 34

- 35: 1 → 35

Now, sum these values:

5 + 7 = 12

12 + 9 = 21

21 + 15+15+15 = 21 + 45 = 66

66 + 16+16 = 66 + 32 = 98

98 + 19 = 117

117 + 20+20 = 117 + 40 = 157

157 + 21 = 178

178 + 25 = 203

203 + 28 = 231

231 + 34 = 265

265 + 35 = 300

Wait, that can't be. Wait, 5+7+9=21; 15*3=45 (21+45=66); 16*2=32 (66+32=98); 19=98+19=117; 20*2=40 (117+40=157); 21=157+21=178; 25=178+25=203; 28=203+28=231; 34=231+34=265; 35=265+35=300. So total sum is 300, and number of data points is 1 (5) +1 (7)+1 (9)+3 (15)+2 (16)+1 (19)+2 (20)+1 (21)+1 (25)+1 (28)+1 (34)+1 (35) = 1+1+1+3+2+1+2+1+1+1+1+1 = 16. So mean is 300 / 16 = 18.75, which rounds to 18.8? Wait, no, 300 divided by 16: 16*18=288, 300-288=12, 12/16=0.75, so 18.75, which is 18.8 when rounded to one decimal place.

Wait, maybe my data points are wrong. Let's re-examine the dot plot:

The first three dots: at 5, 7, 9 (each 2 units apart? No, the x-axis is labeled 5, 10, 15, 20, 25, 30, 35. So between 5 and 10, there are 5 units (5,6,7,8,9,10). So the first three dots are at 5, 7, 9 (each 2 units? No, maybe 5, 6, 7? No, the plot shows three dots before 10. Then a gap until 15, where there are three dots (15,16,17? No, the dot plot has vertical lines, so each line is 1 minute. So:

- 5: 1

- 6: 0

- 7: 1

- 8: 0

- 9: 1

- 10: 0

- 11: 0

- 12: 0

- 13: 0

- 14: 0

- 15: 3 (15,16,17? No, three dots at 15: so 15 appears 3 times)

- 16: 2 (two dots at 16: 16 appears 2 times)

- 17: 0

- 18: 0

- 19: 1 (one dot at 19)

- 20: 2 (two dots at 20)

- 21: 1 (one dot at 21)

- 22: 0

- 23: 0

- 24: 0

- 25: 1 (one dot at 25)

- 26: 0

- 27: 0

- 28: 1 (one dot at 28)

- 29: 0

- 30: 0

- 31: 0

- 32: 0

- 33: 0

- 34: 1 (one dot at 34)

- 35: 1 (one dot at 35)

Now, let's list all data points with their frequencies:

5:1, 7:1, 9:1, 15:3, 16:2, 19:1, 20:2, 21:1, 25:1, 28:1, 34:1, 35:1.

Now, sum all values:

5*1 = 5

7*1 = 7 → total 12

9*1 = 9 → total 21

15*3 = 45 → total 66

16*2 = 32 → total 98

19*1 = 19 → total 117

20*2 = 40 → total 157

21*1 = 21 → total 178

25*1 = 25 → total 203

28*1 = 28 → total 231

34*1 = 34 → total 265

35*1 = 35 → total 300

Number of data points: 1+1+1+3+2+1+2+1+1+1+1+1 = 16

Mean = 300 / 16 = 18.75 ≈ 18.8 (rounded to one decimal place)

Step2: Confirm the calculation

Sum of all data points: 5 + 7 + 9 + 15 + 15 + 15 + 16 + 16 + 19 + 20 + 20 + 21 + 25 + 28 + 34 + 35 = let's add again:

5 + 7 = 12; +9 = 21; +15 = 36; +15 = 51; +15 = 66; +16 = 82; +16 = 98; +19 = 117; +20 = 137; +20 = 157; +21 = 178; +25 = 203; +28 = 231; +34 = 265; +35 = 300. Correct.

Number of data points: 16. So 300 / 16 = 18.75, which is 18.8 when rounded to one decimal place.

Answer:

The mean is $\boxed{18.8}$ (rounded to one decimal place).