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Concept explainers
Commute times: Every morning, Tania leaves for work a few minutes after 7:00 A.M. For eight days, she keeps track of the time she leaves (the number of minutes after 7:00) and the number of minutes it takes her to get to work. Following are the results.
- Construct a
scatterplot of the length of commute (y) versus the time leaving (x). - Compute the least-squares regression line for predicting the length of commute from the time leaving.
- Compute the coefficient of determination.
- Which point is an outlier?
- Remove the outlier and compute the least-squares regression line for predicting the length of commute from the time leaving.
- Is the outlier influential? Explain.
- Compute the coefficient of determination for the data set with the outlier removed. Is the relationship stronger. weaker; or about equally strong without the outlier?
a.
![Check Mark](/static/check-mark.png)
To Graph:a scatter plot using the length of commute time
Explanation of Solution
Given information: T leaves her home every day few minutes after
Graph:The scatter plot shows the number of minutes after
Interpretation:Each of the data in the table contributes an ordered pair of the form (number of minutes after
We use a scatter plot for this example because to understand the relationship between the two variables as ordered pairs it is useful. The points tend to cluster around the straight line. Therefore, we conclude that the variable on the
Now consider the three points
Therefore, we conclude that the variable on the
Because for positive linear relationship the large value of data associates with large values of data in the plot, while for negative linear relationship the large value of data associates with small values of data in the pot. And in this case, it is difficult to say that because the large values of data associate with both small and large and small values of data associate with both small and large at the same time.
Therefore it is good to measure how strong the linear relationship is, to know this we can calculate the correlation coefficient.
b.
![Check Mark](/static/check-mark.png)
To Calculate: the least square regression line
Answer to Problem 12RE
When two variables have a linear relationship, the points on a scatter plot tend to cluster around a straight line called the least square regression line. It is simplified to be
Explanation of Solution
Given information: T leaves her home every day few minutes after
Formulas Used:
Sample mean:
Sample variance:
Correlation Coefficient:
The least-square regression line:
Calculation: Using the below table for calculation.
The sample means and the sample variances can be calculated as shown.
Now, one can use these to calculate the correlation coefficient as shown.
Finally, to calculate the least square regression line as shown, theser can be used.
Where
c.
![Check Mark](/static/check-mark.png)
To Calculate: the coefficient of determination
Answer to Problem 12RE
The coefficient of determination is
Explanation of Solution
Given information: T leaves her home every day few minutes after
Formulas Used:
Sample mean:
Sample variance:
Correlation Coefficient:
Calculation: the correlation coefficient can be calculated as shown by using the formula.
The correlation coefficient
indicates a positive linear association. Here the value of the correlation coefficient close to zero.
Therefore, one can conclude that the positive linear relationship is weak.
Also,
To calculate the coefficient of determination we need to square the correlation coefficient.
Therefore, the coefficient of determination is
d.
![Check Mark](/static/check-mark.png)
To Find: the outlier point
Explanation of Solution
Given information: Tania leaves her home every day few minutes after
Graph: The scatter plot shows the number of minutes after
Interpretation: An outlier is a value that considerably larger or considerably smaller than most of the values in a data set. It may be resulting from an error in the process of sampling.
So in the given data set, an outlier point can be detected in the ordered pair
Because it is much larger than the other ordered pairs.
e.
![Check Mark](/static/check-mark.png)
To Calculate: the least square regression line without the outlier point.
Answer to Problem 12RE
The least square regression line without theoutlier point is
Explanation of Solution
Given information: T leaves her home every day few minutes after
Formulas Used:
Sample mean:
Sample variance:
Correlation Coefficient:
The least square regression line:
Calculation: the sample means and the sample variances can be calculated as shown without the outlier point.
Now, one can use these to calculate the correlation coefficient without the outlier as shown.
Finally, one can use these to calculate the least square regression line as shown.
Where
f.
![Check Mark](/static/check-mark.png)
To Show: the outlier is influential
Answer to Problem 12RE
No, the outlier is not that much influential.
Explanation of Solution
Given information: T leaves her home every day few minutes after
The least-square regression line with the outlier is
The least-square regression line without the outlier is
The two least square regression lines are so much close and have no huge difference.
Therefore we can conclude that the outlier is not that much influential. Here, the outlier cannot be a result of an error. It is just random data measured along with the other data in the sampling process.
g.
![Check Mark](/static/check-mark.png)
To Find: the coefficient of determination without the outlier and discuss its strength.
Answer to Problem 12RE
Coefficient of determination is
Explanation of Solution
Given information: T leaves her home every day few minutes after
Formula Used:Sample mean:
Sample variance:
Correlation Coefficient:
Calculation: to calculate the correlation coefficient without the outlier as shown.
The correlation coefficient
indicates a positive linear association. It is
The value is so close to zero .because of the ten to the power is minus thirty-one.
Therefore, one can conclude that the positive linear relationship is very weak without the outlier.
Also,
One can calculate the coefficient of determination by squaring the correlation coefficient.
Therefore, the coefficient of determination is
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Chapter 4 Solutions
Elementary Statistics (Text Only)
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