The graph shows the number of hours spent studying related to a student score on the unit 4 test

Question

**Test grade/hours studying**
40/1, 50/1, 65 / 2, 75 / 2, 80 / 3, 90 / 5, 80/6, 85/7, 90 / 7, 100 / 7

A. Find the equation for the line of best fit
B. Predict the score for student who does not study at all
C. Create a table of data points

Answer 1

To analyze the given data, we have the following points representing hours studied and the corresponding test scores:

Data Points:

• (1, 40)
• (1, 50)
• (2, 65)
• (2, 75)
• (3, 80)
• (5, 90)
• (6, 80)
• (7, 85)
• (7, 90)
• (7, 100)

A. Finding the Line of Best Fit

To find the line of best fit, we can use least squares regression. Let's summarize the data points.

1. Convert the list of points to usable format:

   • It’s better to aggregate the repeated score for the same number of hours studied.

   • Collecting the average scores for the same hours:

   • (1, 45) -> Average of (40, 50) = 45

   • (2, 70) -> Average of (65, 75) = 70

   • (3, 80)

   • (5, 90)

   • (6, 80)

   • (7, 91.67) -> Average of (85, 90, 100) = 91.67

   So the new set of points is:

   • (1, 45)
   • (2, 70)
   • (3, 80)
   • (5, 90)
   • (6, 80)
   • (7, 91.67)

2. Calculating the Line of Best Fit: We will use the formulas for the slope \( m \) and y-intercept \( b \).

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

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

   Let's calculate the necessary sums.

• \( N = 6 \)
• \( \sum x = 1 + 2 + 3 + 5 + 6 + 7 = 24 \)
• \( \sum y = 45 + 70 + 80 + 90 + 80 + 91.67 = 456.67 \)
• \( \sum xy = (145) + (270) + (380) + (590) + (680) + (791.67) = 45 + 140 + 240 + 450 + 480 + 642.69 = 1998.69 \)
• \( \sum x^2 = 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2 = 1 + 4 + 9 + 25 + 36 + 49 = 124 \)

Plugging in these numbers into the formula for \( m \):

\[ m = \frac{6(1998.69) - (24)(456.67)}{6(124) - (24)^2} = \frac{11992.14 - 10960.08}{744 - 576} = \frac{1032.06}{168} \approx 6.14 \]

Now, substituting \( m \) into the formula for \( b \):

\[ b = \frac{456.67 - 6.14(24)}{6} = \frac{456.67 - 147.36}{6} \approx \frac{309.31}{6} \approx 51.55 \]

So the equation of the line of best fit is:

\[ y \approx 6.14x + 51.55 \]

B. Predicting the Score for a Student Who Does Not Study at All

To predict the score for a student who does not study at all (i.e. \( x = 0 \)):

\[ y \approx 6.14(0) + 51.55 \approx 51.55 \]

Thus, the predicted score for a student who does not study at all is approximately 51.55.

C. Table of Data Points

| Hours Studying | Test Score | |----------------|------------| | 1 | 45 | | 2 | 70 | | 3 | 80 | | 5 | 90 | | 6 | 80 | | 7 | 91.67 |

This completes the analysis of the data points and the line of best fit.

Answer 2

Have the line of best fit in point slope formula form

Answer 3

To express the equation of the line of best fit in point-slope form, we use the following formula:

\[ y - y_1 = m(x - x_1) \]

Where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

From our previous calculations, we found the slope \( m \approx 6.14 \). We can use one of the points from our dataset as \( (x_1, y_1) \). For example, let's use the point \( (1, 45) \).

Plugging in these values into the point-slope formula:

\[ y - 45 = 6.14(x - 1) \]

This is the equation of the line of best fit in point-slope form.

If you need the point-slope form with a different point, you can replace \( (1, 45) \) with any other point from the dataset, such as \( (2, 70) \) or \( (3, 80) \), using the same slope \( m = 6.14 \). For example:

Using the point \( (2, 70) \):

\[ y - 70 = 6.14(x - 2) \]

You can choose any point according to your preference!