The table below gives the number of hours spent unsupervised each day as well as the overall grade averages for seven randomly selected middle school students. Using this data, consider the equation of the regression line, yˆ=b0+b1x
, for predicting the overall grade average for a middle school student based on the number of hours spent unsupervised each day. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Hours Unsupervised	0.5
1.5
2.5
3
4
4.5
5
Overall Grades	97
91
87
77
76
74
71
Table


Step 4 of 6 :  Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.


The statement "Not all points predicted by the linear model fall on the same line" is **true**. 

Here's why:

In regression analysis, the goal is to find the best-fit line that represents the relationship between the independent variable (hours unsupervised) and the dependent variable (overall grade). However, the points from the data do not necessarily lie perfectly on the regression line. The line is a best-fit approximation, meaning some points will be above or below the line, which indicates the residuals (the difference between the observed and predicted values). This discrepancy occurs because of the natural variation in data, measurement errors, and the fact that not all relationships are perfectly linear.

In summary, while the regression line tries to minimize the vertical distance (errors) between the points and the line, it is rare for every data point to fall exactly on the line. Hence, the statement is **true**.

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Would you like to proceed with finding the regression line equation or explore another concept? Let me know!

Here are some follow-up questions that might interest you:
1. How do we calculate the correlation coefficient for this data?
2. What are residuals, and how do we interpret them in regression analysis?
3. How can we calculate the equation of the regression line \( y = b_0 + b_1x \)?
4. How can we assess the statistical significance of the correlation coefficient?
5. What is the impact of outliers on the regression model?

**Tip:** When interpreting a regression line, it’s essential to check for the residuals (differences between observed and predicted values) to evaluate how well the model fits the data.

The table below gives the number of hours spent unsupervised each day as well as the overall grade averages for seven randomly selected middle school students. Using this data, consider the equation of the regression line, yˆ = b0 + b1x, for predicting the overall grade average for a middle school student based on the number of hours spent unsupervised each day. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Hours Unsupervised: [0.5, 1.5, 2.5, 3, 4, 4.5, 5]
Overall Grades: [97, 91, 87, 77, 76, 74, 71]

Step 4 of 6: Determine if the statement 'Not all points predicted by the linear model fall on the same line' is true or false.

Explore how linear regression is used to predict overall grades based on hours spent unsupervised. This problem explains the regression equation, residuals, and correlation coefficient, helping students understand why data points do not always fall perfectly on the best-fit line. Suitable for Grades 8-10.

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