Here is a bivariate data set. y 52.8 36.9 51.5 34.6 48.4 35.7 22.2 54.6 60.5 17 49 39 66.1 24 56.1 37.6 54.2 48.5 50.2 29.9 61.1 41.1 Find the correlation coefficient and report it accurate to three decimal places. 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.
 alt="### Bivariate Data Set and Analysis
Here is a bivariate data set:
| x | y |
|-------|-------|
| 52.8 | 36.9 |
| 51.5 | 34.6 |
| 48.4 | 35.7 |
| 22.2 | 54.6 |
| 60.5 | 17 |
| 49 | 39 |
| 66.1 | 24 |
| 56.1 | 37.6 |
| 54.2 | 48.5 |
| 50.2 | 29.9 |
| 61.1 | 41.1 |
**Instructions:**
1. **Find the Correlation Coefficient**
- Calculate the correlation coefficient (r) for the given data set and report it accurate to three decimal places.
- \( r = \_\_\_\_\_ \)
2. **Proportion of Variation Explained**
- Determine what proportion of the variation in \( y \) can be explained by the variation in the values of \( x \). Report the answer as a percentage accurate to one decimal place.
- \( R^2 = \_\_\_\_\_ \% \)
### Guidance
- **Correlation Coefficient (r):**
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values close to 0 indicate a weak linear relationship.
- **Proportion of Variation Explained (R²):**
\( R^2 \) is the coefficient of determination that indicates the proportion of the variance in the dependent variable (\( y \)) that is predictable from the independent variable (\( x \)). It is expressed as a percentage.
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