Question

the dot plot shows the number of hours, rounded to the nearest half hour, it took alexis to box braid the hair of 30 customers. alexis says that the mean time is greater than 7 hours. is she correct? use the drop-down menus to explain. box braids dot plot with time from 2 to 10 click the arrows to choose an answer from each menu. the data in the dot plot are choose... and have a median of choose... hours. based on the shape of the data, the mean time will be choose... the median time. therefore, alexis is choose... .

Explanation:

Step1: Analyze data symmetry/skewness

The dot plot has fewer dots on the left (lower values: 2, 3, 4) and more on the right around 5 - 9. Wait, actually, let's count the dots. Let's list the number of dots at each time:
- 2: 2 dots
- 3: 2 dots? Wait no, the plot: at 2, 2 dots? Wait the x - axis is 2, 3, 4, 5, 6, 7, 8, 9, 10. Let's count properly. Wait the left side (2,3,4) has few dots, and the main cluster is from 5 - 9, with peak at 7? Wait no, the dots: at 2: 2? Wait no, looking at the plot: 2 has 2 dots? Wait 3 has 2? 4 has 1? Then 5: let's see, the dots at 5: maybe 2? Wait no, let's do it step by step. Wait the total number of customers is 30. Let's count the dots:

- Time = 2: 2 dots
- Time = 3: 2 dots? Wait no, the first two dots at 2, then two at 3? Wait no, the plot: leftmost at 2 (2 dots), 3 (2 dots? Wait no, the first mark is 2, then 3, 4, 5, etc. Wait maybe:

Time 2: 2 dots

Time 3: 2 dots? Wait no, maybe:

Time 2: 2

Time 3: 2

Time 4: 1

Time 5: 4 (since the dots start increasing from 5)

Time 6: 5

Time 7: 6 (peak)

Time 8: 5

Time 9: 4

Time 10: 1? Wait no, total should be 30. Let's check: 2 + 2+1 + 4+5+6+5+4+1 = 2+2=4, +1=5, +4=9, +5=14, +6=20, +5=25, +4=29, +1=30. Yes. So the data has a left - skew? Wait no, left skew is tail on left, right skew is tail on right. Here, the tail is on the left (lower times: 2,3,4 have few dots), so the data is left - skewed? Wait no, left skew means mean < median, right skew mean > median. Wait, actually, when data is skewed left (tail on left), the mean is less than the median. When skewed right (tail on right), mean is greater than median. Wait, in our case, the low values (2,3,4) are the tail (fewer data points), so the distribution is left - skewed? Wait no, let's think again. The majority of data is from 5 - 9, with peak at 7. The left side (2,3,4) has a long tail (few points). So the distribution is left - skewed? Wait, no: left skew is when the mean is pulled down by the low tail, right skew is mean pulled up by high tail. Wait, in this case, the low tail (2,3,4) has small values, so the mean would be less than the median? Wait no, let's calculate median first.

Step2: Calculate median

There are 30 data points. The median is the average of the 15th and 16th values when ordered. Let's order the data:

First, count the cumulative number of dots:

- Time = 2: 2 (cumulative: 2)
- Time = 3: 2 (cumulative: 4)
- Time = 4: 1 (cumulative: 5)
- Time = 5: 4 (cumulative: 9)
- Time = 6: 5 (cumulative: 14)
- Time = 7: 6 (cumulative: 20)
- Time = 8: 5 (cumulative: 25)
- Time = 9: 4 (cumulative: 29)
- Time = 10: 1 (cumulative: 30)

The 15th and 16th values: the 14th value is at time 6 (cumulative 14), so 15th and 16th are at time 7. So median is 7 hours.

Step3: Analyze skewness and mean - median relationship

The data has a left - skew? Wait no, the tail is on the left (low values), so the distribution is left - skewed? Wait, no: left skew (negative skew) has mean < median, right skew (positive skew) has mean > median. Wait, but in our case, the low values (2,3,4) are the tail. Let's think about the mean. The mean is calculated as $\frac{\sum (x\times f(x))}{n}$, where $f(x)$ is the frequency.

Let's calculate the sum:

- Time 2: $2\times2 = 4$
- Time 3: $3\times2 = 6$
- Time 4: $4\times1 = 4$
- Time 5: $5\times4 = 20$
- Time 6: $6\times5 = 30$
- Time 7: $7\times6 = 42$
- Time 8: $8\times5 = 40$
- Time 9: $9\times4 = 36$
- Time 10: $10\times1 = 10$

Sum = 4+6 + 4+20+30+42+40+36+10 = let's calculate step by step:

4+6=10; 10+4=14; 14+20=34; 34+30=64; 64+42=106; 106+40=146; 146+36=182; 182+10=192.

Mean = $\frac{192}{30}=6.4$ hours. Wait, that's less than 7. Wait, but maybe my frequency count is wrong. Wait let's re - count the dots. Let's look at the dot plot again. The plot: at 2: 2 dots, 3: 2 dots, 4: 1 dot, 5: let's see, the dots at 5: maybe 3? Wait no, the original plot: "Box Braids" dot plot. Let's assume the correct frequency:

Wait maybe I miscounted. Let's do it again. Let's look at the vertical dots:

At time 2: 2 dots

Time 3: 2 dots

Time 4: 1 dot

Time 5: 3 dots? No, the peak is at 7. Wait maybe the correct counts are:

Time 2: 2

Time 3: 2

Time 4: 1

Time 5: 4

Time 6: 5

Time 7: 6

Time 8: 5

Time 9: 4

Time 10: 1

Total: 2+2+1+4+5+6+5+4+1 = 30. So sum is 2*2 + 3*2+4*1 +5*4 +6*5 +7*6 +8*5 +9*4 +10*1 = 4 + 6+4 +20 +30 +42 +40 +36 +10 = 192. Mean is 192/30 = 6.4, which is less than 7.

Wait, but the problem says Alexis says mean is greater than 7. So let's re - examine the skewness. Wait, maybe the data is right - skewed? No, the tail is on the left. Wait, maybe my frequency is wrong. Let's check the median again. 30 data points, median is (15th + 16th)/2. The first 2 (time 2) + 2 (time 3) +1 (time 4) +4 (time 5) +5 (time 6) = 2+2+1+4+5 = 14. So 14th data point is at time 6. Then 15th and 16th are at time 7. So median is 7.

Now, for a left - skewed distribution (tail on left), mean < median. So mean is less than 7. Therefore, Alexis is incorrect.

Now, the drop - down menus:

1. The data in the dot plot are skewed left (because the tail is on the left, lower values)
2. Median is 7 hours (as 15th and 16th are at 7)
3. Based on the shape (left - skewed), mean time will be less than the median time.
4. Therefore, Alexis is incorrect.

Answer:

The data in the dot plot are skewed left and have a median of 7 hours. Based on the shape of the data, the mean time will be less than the median time. Therefore, Alexis is incorrect.