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 image
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 more dots on the left (lower times: 2,3,4,5) and tapers on the right. So data is left - skewed (or skewed left), but wait, actually, the left has fewer dots? Wait, no: count dots. Let's count: 2:2, 3:2? Wait, no, the plot: 2 has 2 dots? Wait, no, looking at the plot: 2: 2 dots? Wait, 3: 2? 4:1? 5:4? 6:5? 7:6? 8:5? 9:3? 10:1? Wait, total should be 30. Let's sum: 2 (2) + 3 (2) +4(1)+5(4)+6(5)+7(6)+8(5)+9(3)+10(1) = 2+2+1+4+5+6+5+3+1=29? Wait, maybe my count is wrong. But the key is the distribution: the left (lower values: 2,3,4) have some dots, and the main cluster is around 6 - 7, but there are a few low - value outliers (2,3,4). Wait, no, low - value outliers would pull the mean down? Wait, no: if data is skewed left (tail on left), mean < median; if skewed right (tail on right), mean > median. Wait, here, the left has some dots (2,3,4) which are lower than the main cluster (5 - 9). So the tail is on the left (lower values), so data is left - skewed? Wait, no, tail is the less frequent side. Wait, the main cluster is from 5 - 9, with peak at 7. The left (2,3,4) has fewer dots, so the tail is on the left. So skewed left. But wait, let's find median. There are 30 data points, so median is average of 15th and 16th terms. Let's order the data:

Count cumulative:

2: 2 (cumulative 2)

3: 2 (cumulative 4)

4:1 (cumulative 5)

5:4 (cumulative 9)

6:5 (cumulative 14)

7:6 (cumulative 20)

Ah, so 15th term is in 7 (since 14th is last of 6, 15th is first of 7), 16th term is also in 7. So median is 7 hours.

Now, for skewness: if data is skewed left (tail on left, low values), mean is less than median. But wait, the low values (2,3,4) are pulling the mean down? Wait, no, wait: 2,3,4 are lower than 7. So adding these low values will make the mean less than the median (which is 7). Wait, but Alexis says mean is greater than 7.

Wait, maybe I messed up skewness. Let's re - check: the right side (8,9,10) has some dots, but the left side (2,3,4) has fewer. Wait, the peak is at 7, and the left has a few low values, right has a few high values. Wait, maybe it's approximately symmetric? No, the left has 2 + 2+1 = 5 dots, right has 5 + 3+1 = 9 dots? Wait, no, 8:5, 9:3, 10:1: 5 + 3+1 = 9. Left: 2:2, 3:2, 4:1: 5. Middle: 5:4, 6:5, 7:6: 15. So total 5+15+9 = 29? Maybe a typo, but assume 30. Anyway, median is 7 (since 15th and 16th are in 7). Now, for mean: the low values (2,3,4) are below 7, and high values (8,9,10) are above 7. But how many? Low values: let's say 2 (2s), 2 (3s), 1 (4): total 5 data points with value <7. High values: 5 (8s), 3 (9s), 1 (10s): total 9 data points with value >7. Middle: 5 (5s), 5 (6s), 6 (7s): 16 data points with value ≤7 (wait, 5+5+6 = 16). Wait, no, 5 (5s): 5, 6 (6s):6, 7 (7s):7. So cumulative up to 7: 2 (2) + 2 (3)+1 (4)+4 (5)+5 (6)+6 (7)=2+2+1+4+5+6 = 20. So 20 data points ≤7, 10 data points >7 (since 30 - 20 = 10). The data points >7: 8 (5), 9 (3), 10 (1): total 9? Wait, 30 - 20 = 10, so maybe my count is wrong. Anyway, the key is: the median is 7. Now, the mean: the low - value outliers (2,3,4) are far below 7, and the high - value points (8,9,10) are above 7. But the number of low - value points: let's say 2 (2s: 2*2 = 4), 2 (3s: 3*2 = 6), 1 (4: 4*1 = 4), 4 (5s: 5*4 = 20), 5 (6s: 6*5 = 30), 6 (7s: 7*6 = 42), 5 (8s: 8*5 = 40), 3 (9s: 9*3 = 27), 1 (10: 10*1 = 10). Now sum all: 4+6+4+20+30+42+40+27+10 = let's calculate: 4+6 = 10; 10+4 = 14; 14+20 = 34; 34+30 = 64; 64+42 = 106; 106+40 = 146; 146+27 = 173; 173+10 = 183. Wait, but total data points: 2+2+1+4+5+6+5+3+1 = 29. Oh, so maybe one more dot. Let's assume the total is 30, so maybe one more dot at 7. Then sum would be 183 + 7 = 190. Mean = 190 / 30≈6.33? Wait, that can't be. Wait, I must have messed up the dot count. Let's look at the plot again: the dot plot for Box Braids: at 2: 2 dots, 3: 2 dots, 4:1 dot, 5:4 dots, 6:5 dots, 7:6 dots, 8:5 dots, 9:3 dots, 10:1 dot. Wait, 2+2+1+4+5+6+5+3+1 = 29. So maybe the problem has 29 data points, but it says 30. Anyway, the median: for n = 30 (even), median is (15th + 16th)/2. Let's list the positions:

1 - 2: 2

3 - 4: 3

5: 4

6 - 9: 5

10 - 14: 6

15 - 20: 7

21 - 25: 8

26 - 28: 9

29: 10

So 15th and 16th are both 7, so median is 7.

Now, the data: the left (2,3,4) are low values (below 7), and the right (8,9,10) are high values (above 7). But the number of low - value points (2,3,4) is 2 + 2+1 = 5, and high - value points (8,9,10) is 5 + 3+1 = 9. The low - value points are further from 7 (2 is 5 less, 3 is 4 less, 4 is 3 less) than the high - value points (8 is 1 more, 9 is 2 more, 10 is 3 more). So the pull of the low - value points (which are more distant below 7) will make the mean less than the median (7). Wait, but that contradicts? Wait, no: mean is affected by outliers. If there are low - value outliers, mean < median; if high - value outliers, mean > median. Here, the low - value points (2,3,4) are outliers (since most data is 5 - 9), so mean < median (7). Therefore, Alexis says mean >7, which is incorrect.

Now, the drop - down menus:

1. The data in the dot plot are skewed left (because the tail is on the left, low values) or maybe skewed? Wait, actually, the left has a few dots, right has a few, but the main cluster is at 7. Wait, maybe the data is skewed left? Or maybe the first drop - down is "skewed left" or "symmetric" or "skewed right". Wait, since there are low - value outliers (2,3,4) and the right has some high - value points but the left tail is longer (in terms of distance from median). Wait, no, the left tail (2,3,4) is shorter in number of dots but the values are lower. Anyway, the median is 7. Then, based on shape (skewed left), mean is less than median. So Alexis is incorrect.

So filling the drop - downs:

The data in the dot plot are skewed left (or maybe "skewed" with left tail), 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.

Answer:

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