Problem #3
A group of 23 students were taken as a sample size in the study of IQ level versus Social
Intelligence and were found to be:
# IQ SI # IQ SI # IQ SI
1 95 26 9 106 30 17 95 31
2 102 25 10 105 30 18 95 29
3 110 29 11 101 35 19 98 25
4 99 33 12 99 33 20 108 28
5 98 31 13 89 27 21 122 27
6 87 30 14 90 26 22 118 30
7 88 24 15 90 29 23 117 32
8 92 23 16 100 30

A. Compute the pearson-r correlation coefficient (show complete solution
B. Determine the level of correlation (refer to the correlation scale found in the SAS).
C. Make a scatter diagram (use excel or any online graphing calculator

### A. Compute the Pearson-r Correlation Coefficient

The Pearson correlation coefficient, \( r \), is a measure of the linear relationship between two variables \( X \) (IQ) and \( Y \) (SI). The formula for \( r \) is:

\[
r = \frac{n(\sum{XY}) - (\sum{X})(\sum{Y})}{\sqrt{\left[n\sum{X^2} - (\sum{X})^2\right]\left[n\sum{Y^2} - (\sum{Y})^2\right]}}
\]

Where:
- \( X \) represents IQ scores.
- \( Y \) represents Social Intelligence (SI) scores.
- \( n \) is the number of data points (in this case, \( n = 23 \)).

#### Step 1: Collect the data

The data from the question is given as:

| #   | IQ (X) | SI (Y) |
|-----|--------|--------|
| 1   | 95     | 26     |
| 2   | 102    | 25     |
| 3   | 110    | 29     |
| 4   | 99     | 33     |
| 5   | 98     | 31     |
| 6   | 87     | 30     |
| 7   | 88     | 24     |
| 8   | 92     | 23     |
| 9   | 106    | 30     |
| 10  | 105    | 30     |
| 11  | 101    | 35     |
| 12  | 99     | 33     |
| 13  | 89     | 27     |
| 14  | 90     | 26     |
| 15  | 90     | 29     |
| 16  | 100    | 30     |
| 17  | 95     | 31     |
| 18  | 95     | 29     |
| 19  | 98     | 25     |
| 20  | 108    | 28     |
| 21  | 122    | 27     |
| 22  | 118    | 30     |
| 23  | 117    | 32     |

#### Step 2: Calculate the necessary sums

We will need the following values:
- \( \sum{X} \) = sum of IQ scores
- \( \sum{Y} \) = sum of SI scores
- \( \sum{X^2} \) = sum of squares of IQ scores
- \( \sum{Y^2} \) = sum of squares of SI scores
- \( \sum{XY} \) = sum of products of IQ and SI scores

Let me compute these values.The calculated sums are:
- \( \sum{X} = 2304 \)
- \( \sum{Y} = 663 \)
- \( \sum{X^2} = 232910 \)
- \( \sum{Y^2} = 19321 \)
- \( \sum{XY} = 66590 \)
- \( n = 23 \)

#### Step 3: Substitute into the Pearson-r formula

\[
r = \frac{n(\sum{XY}) - (\sum{X})(\sum{Y})}{\sqrt{\left[n\sum{X^2} - (\sum{X})^2\right]\left[n\sum{Y^2} - (\sum{Y})^2\right]}}
\]

Substitute the calculated values to find \( r \).It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can continue the process by manually calculating the Pearson correlation using the values I provided and the formula.

Here’s the next step:

- **Numerator**: \( n(\sum{XY}) - (\sum{X})(\sum{Y}) \)
- **Denominator**: \( \sqrt{\left[n\sum{X^2} - (\sum{X})^2\right]\left[n\sum{Y^2} - (\sum{Y})^2\right]} \)

Finally, divide the numerator by the denominator to get the Pearson correlation coefficient \( r \).

#### B. Determine the Level of Correlation
Once you have \( r \), refer to the SAS correlation scale:
- \( 0.0 - 0.19 \): Very weak
- \( 0.2 - 0.39 \): Weak
- \( 0.4 - 0.59 \): Moderate
- \( 0.6 - 0.79 \): Strong
- \( 0.8 - 1.0 \): Very strong

#### C. Scatter Diagram
To create a scatter diagram, you can use a tool like Excel:
1. Enter the IQ (X) and SI (Y) values into two columns.
2. Highlight the data and insert a scatter plot.

---

Would you like more details on any of these steps?

Here are 5 questions to explore further:
1. What does the Pearson correlation coefficient indicate about relationships?
2. How is the Pearson correlation coefficient different from Spearman’s rank correlation?
3. Can outliers affect the Pearson correlation coefficient significantly?
4. How would you interpret a negative Pearson correlation?
5. What are the assumptions for using Pearson correlation in a dataset?

**Tip:** When interpreting a correlation, always consider both the strength and the direction (positive or negative).

A group of 23 students were taken as a sample size in the study of IQ level versus Social Intelligence (SI) and were found to be:
# IQ SI # IQ SI # IQ SI
1 95 26 9 106 30 17 95 31
2 102 25 10 105 30 18 95 29
3 110 29 11 101 35 19 98 25
4 99 33 12 99 33 20 108 28
5 98 31 13 89 27 21 122 27
6 87 30 14 90 26 22 118 30
7 88 24 15 90 29 23 117 32
8 92 23 16 100 30

A. Compute the Pearson-r correlation coefficient.
B. Determine the level of correlation (refer to the correlation scale found in the SAS).
C. Make a scatter diagram.

Learn how to compute the Pearson correlation coefficient for a dataset comparing IQ and Social Intelligence among 23 students. This detailed solution includes data collection, calculations, correlation strength interpretation, and a scatter plot. Ideal for advanced high school or undergraduate students studying statistics.

Math Problem Statement

IQ SI # IQ SI # IQ SI

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation