A teacher was conducting a research to see whether there was an association between internal test of students and there exam performances, He collected the test and exam scores at a sample 23 and gra?

ded them on a scatter plot. After analyzing the data, he found a strong positive correlation between the internal test scores and exam performances of the students. This suggests that students who performed well on the internal test also tended to perform well on the exams. The teacher concluded that the internal test scores can be used as a predictor of students' exam performances.

Step 1: Gather data

The teacher collected the test and exam scores of a sample of 23 students.

Step 2: Calculate the correlation coefficient

The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It can range from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

The correlation coefficient between the internal test scores and the exam scores can be calculated using the following formula:

```
r = (Σ(x - x̄)(y - ȳ)) / √(Σ(x - x̄)²Σ(y - ȳ)²)
```

where:

- x is the internal test score
- y is the exam score
- x̄ is the mean of the internal test scores
- ȳ is the mean of the exam scores

Step 3: Interpret the correlation coefficient

The correlation coefficient indicates the strength and direction of the linear relationship between the internal test scores and the exam scores. A positive correlation coefficient indicates that students who scored higher on the internal test also tended to score higher on the exam. A negative correlation coefficient indicates that students who scored higher on the internal test tended to score lower on the exam.

Step 4: Determine the significance of the correlation coefficient

The significance of the correlation coefficient can be determined by calculating the p-value. The p-value is the probability of obtaining a correlation coefficient as large as or larger than the observed correlation coefficient, assuming that there is no real correlation between the two variables.

If the p-value is less than 0.05, then the correlation coefficient is considered to be statistically significant. This means that it is unlikely that the observed correlation coefficient is due to chance.

Step 5: Draw a conclusion

Based on the correlation coefficient and the p-value, the teacher can draw a conclusion about the relationship between internal test scores and exam performances. If the correlation coefficient is positive and statistically significant, then the teacher can conclude that there is a positive association between internal test scores and exam performances. This means that students who score higher on the internal test are more likely to score higher on the exam.