Here is a bivariate data set. y 45.6 79.6 74.1 49 51.8 80.3 50.4 67.9 49.2 83.3 55.9 66.4 42.1 92.7 39.5 84.3 96.3 28.6 Find the correlation coefficient and report it accurate to three decimal places. r = What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. R2 =
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.
![### Bivariate Data Analysis
#### Data Set
We have a bivariate data set represented in the table below with variables \( x \) and \( y \).
| x | y |
|------|------|
| 45.6 | 79.6 |
| 74.1 | 49.0 |
| 51.8 | 80.3 |
| 50.4 | 67.9 |
| 49.2 | 83.3 |
| 55.9 | 66.4 |
| 42.1 | 92.7 |
| 39.5 | 84.3 |
| 96.3 | 28.6 |
#### Instructions
1. **Correlation Coefficient:** Calculate the correlation coefficient (\( r \)) and report it to three decimal places.
\[
r = \_\_\_\_\_
\]
2. **Proportion of Variation Explained ( \( R^2 \)):** Calculate the proportion of the variation in \( y \) that can be explained by the variation in the values of \( x \). Report the answer as a percentage accurate to one decimal place.
\[
R^2 = \_\_\_\_\_ \%
\]
### Detailed Explanation of Concepts
1. **Correlation Coefficient ( \( r \) ):**
- The correlation coefficient measures the strength and direction of the linear relationship between two quantitative variables.
- It ranges from -1 to 1, where:
- 1 indicates a perfect positive linear relationship,
- -1 indicates a perfect negative linear relationship,
- 0 indicates no linear relationship.
2. **Coefficient of Determination ( \( R^2 \) ):**
- The \( R^2 \) value, also known as the coefficient of determination, represents the proportion of the variance in the dependent variable (\( y \)) that can be explained by the independent variable (\( x \)).
- It is calculated as the square of the correlation coefficient (\( r^2 \)) and expressed as a percentage.
- For example, an \( R^2 \) value of 0.64 (or 64%) indicates that 64% of the variance in \( y \) can be explained by the variance in \( x \).
Complete the calculations to find the values of \( r \) and \( R^2 \) as per the instructions provided](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa218d9b0-33cd-49fb-8966-615774819802%2F9f9cc288-4e6e-40c4-8515-73a8523d9340%2Fxkdmqsh_processed.png&w=3840&q=75)
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