Question

Hello this is a multi step question and I am struggling to help my son with this. It is 1 of 3 so hoping to get guidance with this first one to be able to know how to apply it to the others in his activities. Thank you as I know this is multiple steps and time consuming. The help is greatly appreciated as a parent.

Answer 1

In the first part of this problem, we must compute some statistic variables of two distributions:

0. the mean value,

,

1. the median,

,

2. the standard deviation.

,

3. the interquartile range.

1. The mean of a data set is the sum of all the data divided by the count n:

`\mu=(x_1+x_2+\cdots+x_n)/(n)\text{.}`

2. The median is the data value separating the upper half of a data set from the lower half, it is computed following these steps:

• arrange data values from lowest to the highest value,

,

• the median is the data value in the middle of the set

,

• if there are 2 data values in the middle the median is the mean of those 2 values.

3. The standard deviation for a sample data set is given by the following formula:

`\sigma=\sqrt[]{\frac{(x_1-\mu)^2+(_{}x_2-\mu)^2+\cdots+(x_n-\mu)^2}{n-1}_{}}\text{.}`

4. The interquartile range (IQR) is given by:

`\text{IQR}=Q_3-Q_1\text{.}`

Where Q_1 and Q_3 are the first and third quartiles. The lowest quartile (Q1) covers the smallest quarter of values in your dataset.

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Using the definitions above, we compute the mean, the median and the standard deviation for the samples taken by Manuel and Gretchen.

Manuel's sample

• Sample = {3, 6, 8, 11, 12, 8, 6, 3, 10, 5, 14, 9, 7, 10, 8}

,

• Count = 15

1. Mean

Using the formula above, we get:

`\mu=(120)/(5)=8.`

2. Median

We order the data set:

`3,3,5,6,6,7,8,(8),8,9,10,10,11,12,14.`

From the ordered data set, we see that the central number 8 divides the data set into two equal parts.

So the median of this sample is:

`\bar{x}=8.`

3. Standard deviation

Using the formula above, we get:

`\sigma=\sqrt[]{(138)/(15-1)}\cong3.14.`

4. Interquartile range

Dividing the data sample into quartiles, we have:

`3,3,5,6|6,7,8|8|8,9,10|10,11,12,14.`

We have:

• Q_1 = 6,

,

• Q_3 = 10.

So the interquartile range is:

`\text{IQR }=Q_3-Q_1=10-6=4.`

Gretchen's sample

• Sample = {22, 4, 7, 8, 12, 15, 10, 7, 9, 6, 13, 3, 8, 10, 10}

,

• Count = 15

1. Mean

`\mu=(144)/(15)=9.6.`

2. Median

We order the data set:

`3,4,6,7,7,8,8,(9),10,10,10,12,13,15,22.`

From the ordered data set, we see that the central number 8 divides the data set into two equal parts.

So the median of this sample is:

`\bar{x}=9.`

3. Standard deviation

`\sigma=\sqrt[]{(307.6)/(15-1)}\cong4.69.`

4. Interquartile range

Dividing the data sample into quartiles, we have:

`3,4,6,7|7,8,8|9|10,10,10|12,13,15,22.`

We have:

• Q_1 = 7,

,

• Q_3 = 12.

So the interquartile range is:

`\text{IQR }=Q_3-Q_1=12-7=5.`

Answers

Manuel's sample

0. Mean = 8

,

1. Median = 8

,

2. Standard deviation ≅ 3.14

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3. Interquartile range = 4

Gretchen's sample

0. Mean = 9.6

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1. Median = 9

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2. Standard deviation ≅ 4.69

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3. Interquartile range = 5