Understanding Basic Statistics
Understanding Basic Statistics
8th Edition
ISBN: 9781337558075
Author: Charles Henry Brase, Corrinne Pellillo Brase
Publisher: Cengage Learning
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Chapter 4.1, Problem 23P

(a)

To determine

The scatter plot, whether the provided values of x, y, x2 ,y2, xy are correct or not, and the correlation coefficient.

(a)

Expert Solution
Check Mark

Answer to Problem 23P

Solution: The provided values, that is, x=154, y=249, x2=3712, y2=9959 xy=6067 are correct and the value of r is 0.991.

Explanation of Solution

Given: The provided table consists of values of x and y, where x represents the average annual hours spent by a person in traffic delay, y represents the average annual gallons of fuel wasted per person due to traffic delay. The data consists of 8 data pairs, thus n is 8.

Calculation: Follow the steps given below in MS Excel to obtain the scatter plot of the data.

Step 1: Enter the data into an MS Excel sheet. The screenshot is given below.

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  1

Step 2: Select the data and click on ‘Insert’. Go to charts and select the chart type ‘Scatter’.

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  2

Step 3: Select the first plot and then click ‘add chart element’ provided in the left corner of the menu bar. Insert the ‘Axis titles’ and ‘Chart title’. The scatter plot for the provided data is shown below:

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  3

To calculate x, y, x2 ,y2 and xy, it is easy to form the data in a table of five columns. The table is given below:

x y x2 y2 xy
28 48 784 2304 1344
5 3 25 9 15
20 34 400 1156 680
35 55 1225 3025 1925
20 34 400 1156 680
23 38 529 1444 874
18 28 324 784 504
5 9 25 81 45
x=154 y=249 x2=3712 y2=9959 xy=6067

The provided values, x=154, y=249, x2=3712, y2=9959 and xy=6067 have been verified.

Now, the value of r can be calculated by using the formula below:

r=nxy-(x)(y)nx2(x)2ny2(y)2

Substituting the values in the above formula. Thus:

r=8(6067)(154)(249)(8)(3712)(154)2(8)(9959)(249)20.991

Therefore, the correlation coefficient is 0.991.

(b)

To determine

The averages x¯,y¯ and the standard deviations sx,sy for both the data sets, the comparison between the standard deviations of both the samples, and the reason behind the tendency of an increase in the value of r for smaller standard deviations sx and sy.

(b)

Expert Solution
Check Mark

Answer to Problem 23P

Solution: The values for data set 1 are x¯=19.25,y¯=31.13,sx10.33 and sy=17.76.

The values for data set 2 are x¯=20.13,y¯=31.87,sx13.84 and sy=25.18.

Explanation of Solution

Given: The provided table consists of values of x and y, where x represents the average annual hours spent by a person in traffic delay, y represents the average annual gallons of fuel wasted per person due to traffic delay.

The second table consists of x and y values where, x represent the annual hours lost by a person spent in traffic delay, y represents the annual gallons of fuel wasted by that person in traffic delay.

The data sets consist of 8 data pairs, thus n is 8 for both the data sets.

The provided values of data set 1 are, x=154, y=249, x2=3712, y2=9959 xy=6067.

The provided values of data set 2 are, x=161, y=255, x2=4583, y2=12565 xy=7071.

Calculation:

The value of x¯ for data set 1 can be calculated as follows:

x¯=xn=1548=19.25

The value of y¯ for data set 1 can be calculated as follows:

y¯=yn=2498=31.125

The standard deviation of x for data set 1 can be calculated as,

sx=x2(x)2nn1=3712154288110.33

The standard deviation of y for data set 1 can be calculated as,

sy=y2(y)2nn1=9959249288117.76

The value of x¯ for data set 2 can be calculated as follows:

x¯=xn=1618=20.13

The value of y¯ for data set 2 can be calculated as follows:

y¯=yn=2558=31.87

The standard deviation of x for data set 2 can be calculated as,

sx=x2(x)2nn1=4583161288113.84

The standard deviation of y for data set 2 can be calculated as,

sy=y2(y)2nn1=12565255288125.18

For the second data set, that is, for the variables based on single individuals, the standard deviations sx and sy are larger.

The values sx and sy are in the denominator in the formula for calculating r. Dividing by smaller values of sx and sy tends to increase the value of r.

(c)

To determine

The scatter plot, whether the provided values of x, y, x2 ,y2, xy are correct or not, and the correlation coefficient.

(c)

Expert Solution
Check Mark

Answer to Problem 23P

Solution: The provided values, that is, x=161, y=255, x2=4583, y2=12565 xy=7071 are correct and the value of r is 0.794.

Explanation of Solution

The provided table consists of values of x and y, where x represents the average annual hours spent by a person in traffic delay, y represents the average annual gallons of fuel wasted per person due to traffic delay.

The data sets consist of 8 data pairs, thus n is 8.

Calculation: Follow the steps given below in MS Excel to obtain the scatter plot of the data.

Step 1: Enter the data into an MS Excel sheet. The screenshot is given below.

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  4

Step 2: Select the data and click on ‘Insert’. Go to charts and select the chart type ‘Scatter’.

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  5

Step 3: Select the first plot and then click ‘add chart element’ provided in the left corner of the menu bar. Insert the ‘Axis titles’ and ‘Chart title’. The scatter plot for the provided data is shown below:

Understanding Basic Statistics, Chapter 4.1, Problem 23P , additional homework tip  6

Calculation: The calculation for x, y, x2 ,y2 and xy is shown below;

x y x2 y2 xy
20 60 400 3600 1200
4 8 16 64 32
18 12 324 144 216
42 50 1764 2500 2100
15 21 225 441 315
25 30 625 900 750
2 4 4 16 8
35 70 1225 4900 2450
x=161 y=255 x2=4583 y2=12565 xy=7071

The provided values, x=161, y=255, x2=4583, y2=12565 and xy=7071 have been verified.

Now, the value of r can be calculated by using the formula below:

r=nxy-(x)(y)nx2(x)2ny2(y)2

Substituting the values in the above formula. Thus:

r=8(7071)(161)(255)(8)(4583)(161)2(8)(12565)(255)20.794

Therefore, the correlation coefficient is 0.794.

(d)

To determine

Comparison between the values of r that are calculated in part (a) and part (c), whether the data for average have a higher correlation coefficient than the data for individual measurement or not, and the reason for it.

(d)

Expert Solution
Check Mark

Answer to Problem 23P

Solution: Yes, the data for average has a higher correlation coefficient than the data for individual measurement because, according to the central limit theorem, the standard deviation of averages will be smaller than the standard deviation of individual values.

Explanation of Solution

Given: The values of correlation coefficient from part (a) and part (b) are 0.991 and 0.794, respectively.

It can be seen that 0.991>0.794. The data for average has a higher correlation coefficient than the data for individual measurement. This is because the standard deviation for the average is smaller than the standard deviation for individual measurements.

According to the central limit theorem, the standard deviation is smaller for the x¯ distribution than the corresponding x distribution.

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Chapter 4 Solutions

Understanding Basic Statistics

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