Answer :
First, let's tabulate the given data for clarity:
| High School Average | College Average |
|---------------------|-----------------|
| 83 | 77 |
| 75 | 70 |
| 98 | 93 |
| 81 | 77 |
| 77 | 79 |
| 68 | 65 |
| 92 | 87 |
| 70 | 65 |
| 86 | 88 |
| 75 | 70 |
| 89 | 85 |
### Step 1: Create a Scatter Plot
To create a scatter plot, we will plot the High School Averages on the x-axis and the corresponding College Averages on the y-axis. Each pair of values (High School Average, College Average) will be represented as a point on the graph.
1. Plotting Points: Mark the points (83, 77), (75, 70), (98, 93), (81, 77), (77, 79), (68, 65), (92, 87), (70, 65), (86, 88), (75, 70), and (89, 85) on the graph.
2. Axes: Label the x-axis as "High School Averages" and the y-axis as "College Averages".
3. Title: Give the graph a title, such as "Scatter Plot of High School and College Averages".
Here is an illustration of the scatter plot axes and points:
```
100 +------------------------------------------------+
| |
| |
90 + × +
| |
| × |
| × |
College+ × × × +
| |
80 + × × × +
| |
| |
| |
70 + × × × × +
| |
| |
| |
60 + +
| |
| |
50 +------------------------------------------------+
50 60 70 80 90 100
High School Averages
```
### Step 2: Determine Correlation
To determine the correlation between the High School Averages and College Averages, calculate the correlation coefficient. The correlation coefficient, usually denoted as [tex]\( r \)[/tex], quantifies the degree to which two variables are related.
#### Steps to Calculate the Correlation Coefficient [tex]\( r \)[/tex]:
1. Find the mean of each set of averages:
- Mean of High School Averages:
[tex]\[ \bar{X} = \frac{83 + 75 + 98 + 81 + 77 + 68 + 92 + 70 + 86 + 75 + 89}{11} = \frac{894}{11} = 81.27 \][/tex]
- Mean of College Averages:
[tex]\[ \bar{Y} = \frac{77 + 70 + 93 + 77 + 79 + 65 + 87 + 65 + 88 + 70 + 85}{11} = \frac{856}{11} = 77.82 \][/tex]
2. Calculate the deviations from the mean for each paired value:
- For example, for the first pair (83, 77):
[tex]\[ (83 - 81.27) = 1.73 \][/tex]
[tex]\[ (77 - 77.82) = -0.82 \][/tex]
3. Calculate the product of the deviations for each pair and sum these products:
[tex]\[ \sum (X_i - \bar{X})(Y_i - \bar{Y}) \][/tex]
4. Calculate the squared deviations from the mean for each set and sum them:
[tex]\[ \sum (X_i - \bar{X})^2 \quad \text{and} \quad \sum (Y_i - \bar{Y})^2 \][/tex]
5. Compute the correlation coefficient using the formula:
[tex]\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} \][/tex]
### Explanation of Result:
- If [tex]\( r \)[/tex] is close to 1, it indicates a strong positive correlation.
- If [tex]\( r \)[/tex] is close to 0, it indicates no correlation.
- If [tex]\( r \)[/tex] is close to -1, it indicates a strong negative correlation.
Given the positive pattern in the scatter plot, visually we can anticipate a positive correlation. Indeed, if calculated correctly, the [tex]\( r \)[/tex] value should be positive and relatively close to 1, indicating a positive correlation between high school and college averages.
| High School Average | College Average |
|---------------------|-----------------|
| 83 | 77 |
| 75 | 70 |
| 98 | 93 |
| 81 | 77 |
| 77 | 79 |
| 68 | 65 |
| 92 | 87 |
| 70 | 65 |
| 86 | 88 |
| 75 | 70 |
| 89 | 85 |
### Step 1: Create a Scatter Plot
To create a scatter plot, we will plot the High School Averages on the x-axis and the corresponding College Averages on the y-axis. Each pair of values (High School Average, College Average) will be represented as a point on the graph.
1. Plotting Points: Mark the points (83, 77), (75, 70), (98, 93), (81, 77), (77, 79), (68, 65), (92, 87), (70, 65), (86, 88), (75, 70), and (89, 85) on the graph.
2. Axes: Label the x-axis as "High School Averages" and the y-axis as "College Averages".
3. Title: Give the graph a title, such as "Scatter Plot of High School and College Averages".
Here is an illustration of the scatter plot axes and points:
```
100 +------------------------------------------------+
| |
| |
90 + × +
| |
| × |
| × |
College+ × × × +
| |
80 + × × × +
| |
| |
| |
70 + × × × × +
| |
| |
| |
60 + +
| |
| |
50 +------------------------------------------------+
50 60 70 80 90 100
High School Averages
```
### Step 2: Determine Correlation
To determine the correlation between the High School Averages and College Averages, calculate the correlation coefficient. The correlation coefficient, usually denoted as [tex]\( r \)[/tex], quantifies the degree to which two variables are related.
#### Steps to Calculate the Correlation Coefficient [tex]\( r \)[/tex]:
1. Find the mean of each set of averages:
- Mean of High School Averages:
[tex]\[ \bar{X} = \frac{83 + 75 + 98 + 81 + 77 + 68 + 92 + 70 + 86 + 75 + 89}{11} = \frac{894}{11} = 81.27 \][/tex]
- Mean of College Averages:
[tex]\[ \bar{Y} = \frac{77 + 70 + 93 + 77 + 79 + 65 + 87 + 65 + 88 + 70 + 85}{11} = \frac{856}{11} = 77.82 \][/tex]
2. Calculate the deviations from the mean for each paired value:
- For example, for the first pair (83, 77):
[tex]\[ (83 - 81.27) = 1.73 \][/tex]
[tex]\[ (77 - 77.82) = -0.82 \][/tex]
3. Calculate the product of the deviations for each pair and sum these products:
[tex]\[ \sum (X_i - \bar{X})(Y_i - \bar{Y}) \][/tex]
4. Calculate the squared deviations from the mean for each set and sum them:
[tex]\[ \sum (X_i - \bar{X})^2 \quad \text{and} \quad \sum (Y_i - \bar{Y})^2 \][/tex]
5. Compute the correlation coefficient using the formula:
[tex]\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} \][/tex]
### Explanation of Result:
- If [tex]\( r \)[/tex] is close to 1, it indicates a strong positive correlation.
- If [tex]\( r \)[/tex] is close to 0, it indicates no correlation.
- If [tex]\( r \)[/tex] is close to -1, it indicates a strong negative correlation.
Given the positive pattern in the scatter plot, visually we can anticipate a positive correlation. Indeed, if calculated correctly, the [tex]\( r \)[/tex] value should be positive and relatively close to 1, indicating a positive correlation between high school and college averages.
