Based on the data shown below, calculate the correlation coefficient (rounded to three decimal places) y 4 23.55 5 20.78 19.91 7 19.04 8 16.07 9. 15.6
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Calculating the Correlation Coefficient
**Given Data:**
| x | y |
|---|------|
| 4 | 23.55|
| 5 | 20.78|
| 6 | 19.91|
| 7 | 19.04|
| 8 | 16.07|
| 9 | 15.6 |
**Task:** Based on the data shown above, calculate the correlation coefficient (rounded to three decimal places).
#### Explanation:
The table presents pairs of values for two variables \( x \) and \( y \). The goal is to determine the correlation coefficient, which indicates the strength and direction of the linear relationship between these two variables.
1. **Data Representation:**
- The table displays two columns, where the first column is labeled \( x \) and the second column is labeled \( y \).
- Each row corresponds to a pair of \( x \) and \( y \) values.
2. **Calculation:**
- Use the Pearson correlation coefficient formula to calculate the correlation coefficient \( r \):
\[
r = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}}
\]
where:
- \( n \) is the number of pairs of data
- \( \sum{xy} \) is the sum of the product of paired scores
- \( \sum{x} \) is the sum of \( x \) scores
- \( \sum{y} \) is the sum of \( y \) is the sum of \( y \) scores
- \( \sum{x^2} \) is the sum of the squares of \( x \) scores
- \( \sum{y^2} \) is the sum of the squares of \( y \) scores
3. **Interpretation:**
- The correlation coefficient \( r \) value ranges from -1 to 1.
- A value close to 1 indicates a strong positive linear relationship.
- A value close to -1 indicates a strong negative linear relationship.
- A value around 0 suggests no linear relationship.
By following these steps, you will be able to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcebbde0b-56df-456f-b45f-057baf1d3a0f%2Fcdc44630-a9c4-41c4-ab26-6087a72d1bee%2F0voe89_processed.png&w=3840&q=75)
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