Question

Please help!

2. A football coach recorded his team’s game scores over a football season. The scores are 21, 45, 21, 14, 21, 28, 24, 14, 24, 28.
(a) Find the mean absolute deviation of the data. Round to the nearest tenth.
(b) Interpret the mean absolute deviation of the data within the context of the problem.
(c) Explain how removing the outlier, 45, affects the mean absolute deviation.


4. Two random samples are shown from two different populations. Sample 1 represents the monthly cost of living for a single person. Sample 2 represents the monthly cost of living for a married couple with one child.
Sample 1: {$1800, $2000, $2200, $2300, $2500, $3000}
Sample 2: {$5400, $5500, $5600, $5600, $5800, $6000}
(a) Calculate the mean and the median of each population.
(b) Draw a conclusion about the populations based on the measures of center.

Answer 1

2a. 5.8

2b. On average, a score will be 5.8 points away from the average score.

2c. The score will be about 2 points nearer to the average.

4a1. Sample 1 Mean: $2300; Sample 1 Median: $2250.

4a2. Sample 2 Mean: $5650; Sample 2 Median: $5600.

4b. The monthly cost of living is more expensive for a married couple w/ 1 child than for a single person.

Explanation:

2a./2b.

In order to find the mean absolute deviation, we need to first find the mean.

(21 + 45 + 21 + 14 + 21 + 28 + 24 + 14 + 24 + 28)/10

(66 + 35 + 49 + 38 + 52)/10

(101 + 87 + 52)/10

(101 + 139)/10

240/10

The mean of this set is 24.

Now for the deviation.

• 21 - 24 = -3
• 45 - 24 = 21
• 14 - 24 = -10
• 28 - 24 = 4
• 24 - 24 = 0

For the mean absolute deviation, divide the combined value of the 10 deviation values from before (in absolute value) by 10.

(|-3| + 21 + |-3| + |-10| + |-3| + 4 + |-10| + 4)/10

(3 + 25 + 3 + 10 + 3 + 10 + 4)/10

(28 + 13 + 13 + 4)/10

(41 + 17)/10

58/10

The mean absolute deviation is 5.8. That interpreted would be like this: A score, on average, is 5.8 points away from the average score.

2c.

What I'll do here is the following: subtract 45 from 240, and then subtract 21 from the mean deviations. In both cases, I will now divide by 9.

(240 - 45)/9

195/9

65/3

The mean of the set (without 45) is now 21 2/3.

(58 - 21)/9

37/9

The new mean absolute deviation is 4 1/9, which rounds to 4.1. That's 1.7 points nearer to the average than originally.

So, without 45 as one of the scores, the M.A.D. decreases by about 2. That interpreted would be like this: The scores are about 2 points nearer to the average than before.

4a./4b.

First, I'll find the mean for sample one.

(1800 + 2000 + 2200 + 2300 + 2500 + 3000)/6

(3800 + 4500 + 5500)/6

(8300 + 5500)/6

13800/6

The mean for sample one is $2,300/month.

Now for the median for sample one. Since there are 6 values, add the two middle values and divide by 2.

(2200 + 2300)/2

4500/2

The median for sample one is $2,250/month.

Now for the mean & median for sample two.

Mean:

(5400 + 5500 + 5600 + 5600 + 5800 + 6000)/6

(10900 + 11200 + 11800)/6

(22100 + 11800)/6

33900/6

The mean for sample two is $5,650/month.

Median:

Again, there are 6 values. However, this time they are the same, so there is no need to divide. The median for sample two is $5,600/month.

Looking at the data, the mean & median for sample was larger than sample one. Thus, I can conclude the following: The monthly cost of living is more expensive for a married couple w/ 1 child than for a single person.