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Concept explainers
(a)
Construct a
Verify the values of
Find the value of r.
(a)
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Answer to Problem 23P
The scatter diagram for data is,
The value of r is 0.991.
Explanation of Solution
Calculation:
The formula for
In the formula, n is the
The variable x denotes average annual hours per person spent in traffic delays and y denotes the average annual gallons of fuel wasted per person in traffic delays.
Step by step procedure to obtain scatter plot using MINITAB software is given below:
- Choose Graph > Scatterplot.
- Choose Simple. Click OK.
- In Y variables, enter the column of x.
- In X variables, enter the column of y.
- Click OK.
The values are verified in the table below,
x | y | 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 |
Hence, the values are verified.
The number of data pairs are
Hence, the value of r is 0.991.
(b)
Find the value of
Find the value of
Find the value of
Find the value of
Identify the set in which standard deviations for x and y is larger.
Explain why standard deviations increase the value of r.
(b)
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Answer to Problem 23P
The value of
The value of
The value of
The value of
The value of
The value of
The value of
The value of
The standard deviation for variables based on single individuals is larger.
The values of
Explanation of Solution
Calculation:
Mean and standard deviation for variables based on averages:
Step by step procedure to obtain mean and standard deviation using MINITAB software is given as,
- Choose Stat > Basic Statistics > Display
Descriptive Statistics . - In Variables enter the column x, y.
- Choose option statistics, and select Mean, standard deviation.
- Click OK.
Output using MINITAB software is given below:
From MINITAB output, the mean of x is 19.25, the mean of y is 31.13, and standard deviation of x is 10.33, the standard deviation of y is 17.76.
Hence, the value of
Mean and standard deviation for variables based on single individuals:
Step by step procedure to obtain mean and standard deviation using MINITAB software is given as,
- Choose Stat > Basic Statistics > Display Descriptive Statistics.
- In Variables enter the column x, y.
- Choose option statistics, and select Mean, standard deviation.
- Click OK.
Output using MINITAB software is given below:
From MINITAB output, the mean of x is 20.13, the mean of y is 31.88, and standard deviation of x is 13.85, the standard deviation of y is 25.18.
Hence, the value of
It can be observed that value of
The formula for correlation coefficient r in equation (1) is,
In the formula the values of
(c)
Construct a scatter diagram for the second set of data.
Verify the values of
Find the value of r.
(c)
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Answer to Problem 23P
The scatter diagram for second set of data is,
The value of r is 0.794.
Explanation of Solution
Calculation:
The variable x denotes average annual number of hours lost for the person and y denotes the average annual gallons of fuel wasted for the person.
Step by step procedure to obtain scatter plot using MINITAB software is given below:
- Choose Graph > Scatterplot.
- Choose Simple. Click OK.
- In Y variables, enter the column of x.
- In X variables, enter the column of y.
- Click OK.
The values are verified in the table below,
x | y | 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 |
Hence, the values are verified.
The number of data pairs are
Hence, the value of r is 0.794.
(d)
Identify the data for averages have a higher
Mention the reasons why hours lost per individual and fuel wasted per individual might vary more than the same quantities averaged over all the people in a city.
(d)
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Explanation of Solution
From part (a), the value of r for averages is 0.991. From part (c), the value of r for single observations is 0.794.
It can be observed that, the correlation coefficient for data of averages is larger than correlation coefficient for data of single observations.
The values of standard deviations for variables x and y based on data of averages are smaller when compared to the values of standard deviations for variables x and y based on data of single observations. The smaller standard deviation results in larger value of r, giving a higher correlation coefficient for data of averages.
Also, based on the central limit theorem the distribution of x distribution has the larger standard deviation when compared to
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Chapter 9 Solutions
Understandable Statistics: Concepts and Methods
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