https://file.aiquickdraw.com/mathbot/file/1724450265997vzla62gt.jpg

The new image provides the correct table for the linear regression problem. Let's perform the calculations again.

### Step 1: Data from the Table
The table now shows:
- Time Spent Studying (hours): 2, 4, 6, 8
- Quiz Score (%): 65, 80, 85, 90

### Step 2: Calculate the Slope (m) and Y-intercept (b)
We will use the same formulas as before:

\[
m = \frac{N\sum{(xy)} - \sum{x}\sum{y}}{N\sum{x^2} - (\sum{x})^2}
\]

\[
b = \frac{\sum{y} - m\sum{x}}{N}
\]

Where:
- \( N \) is the number of data points (4 in this case),
- \( x \) and \( y \) are the individual data points.

### Step 3: Substitute the Values
Substituting the values from the data into these formulas:

- \( \sum{x} = 2 + 4 + 6 + 8 = 20 \)
- \( \sum{y} = 65 + 80 + 85 + 90 = 320 \)
- \( \sum{xy} = (2 \times 65) + (4 \times 80) + (6 \times 85) + (8 \times 90) = 130 + 320 + 510 + 720 = 1680 \)
- \( \sum{x^2} = 2^2 + 4^2 + 6^2 + 8^2 = 4 + 16 + 36 + 64 = 120 \)

### Step 4: Calculate the Slope (m):
\[
m = \frac{(4 \times 1680) - (20 \times 320)}{(4 \times 120) - (20^2)}
\]
\[
m = \frac{6720 - 6400}{480 - 400} = \frac{320}{80} = 4
\]

### Step 5: Calculate the Y-intercept (b):
\[
b = \frac{320 - (4 \times 20)}{4} = \frac{320 - 80}{4} = \frac{240}{4} = 60
\]

### Step 6: Write the Regression Equation
The linear regression equation based on this data is:
\[
y = 4x + 60
\]

### Conclusion
Now let's compare this result with the answer choices in the first image:
- \( y = 0.51x + 61 \)
- \( y = 8.7x + 42.19 \)
- \( y = 5.1x + 65.51 \)
- \( y = 6.2x + 37 \)

The equation we calculated, \( y = 4x + 60 \), does not exactly match any of the provided options. However, based on the slope and intercept, there is no exact match, indicating either a different data set was expected or the answer options were provided with incorrect or approximate values.

### Follow-Up
Would you like to discuss the method further or clarify any steps? 

Here are 5 related questions to expand your understanding:
1. How does changing the number of data points affect the slope and intercept in a linear regression?
2. What are the assumptions behind linear regression, and why are they important?
3. How would you calculate the residuals for this model?
4. What is the impact of outliers on the linear regression equation?
5. How can you use software tools to perform linear regression more efficiently?

**Tip:** Always verify the integrity of your data before performing statistical analysis to ensure accuracy in your calculations and interpretations.

Learn how to calculate linear regression using specific data points for time spent studying and quiz scores. This detailed guide covers the slope and y-intercept formulas, providing a comprehensive understanding of linear regression in statistics.

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