A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study) to be used for predicting test score.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
![**Title: Analyzing Spanish Language Proficiency Using Linear Regression**
**Introduction:**
A study was conducted to examine the relationship between the number of years students have studied Spanish and their scores on a Spanish language proficiency test. A linear regression analysis was performed using this paired data.
**Description of the Analysis:**
A computer program applied the least squares method to fit a linear regression line to the data. The resultant equation and statistical measures are as follows:
**Regression Equation:**
\[ y = mx + b \]
Where:
- \( m \) is the slope of the regression line
- \( b \) is the y-intercept
**Statistics:**
- \( r^2 = 0.83 \): This is the coefficient of determination, indicating that 83% of the variance in the test scores can be explained by the number of years of study.
- \( r = 0.91 \): This is the correlation coefficient, suggesting a strong positive relationship between years of study and test scores.
**Parameters of the Regression Line:**
- Slope (\( m \)): 10.90
- Y-intercept (\( b \)): 31.55
**Prediction:**
The program also calculated a predicted test score for students who have studied Spanish for 2 years.
**Conclusion:**
The analysis affirms that there is a strong linear relationship between the duration of Spanish study and proficiency test performance, with significant predictive value as indicated by high \( r^2 \) and \( r \) values. This model can be utilized for forecasting student performance based on study duration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f35f376-ff20-435c-a0e5-f699a7fdde3b%2F61747d3b-87d5-46a4-9679-2089e6767a6f%2Ftw5lxkm_processed.jpeg&w=3840&q=75)
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