To calculate the estimated intercept ($b_0$) of the regression line ($y = b_0 + b_1x$), you need the following steps:

1. Obtain the slope ($b_1$):
   $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

2. Find the intercept ($b_0$):
   After finding $b_1$, the intercept $b_0$ can be computed as: $b_0 = \bar{y} - b_1 \bar{x}$ where:

   • $\bar{x}$ is the mean of the $x$ values (Hours Unsupervised).
   • $\bar{y}$ is the mean of the $y$ values (Overall Grades).

Step-by-Step Solution:

Given data:

• $x$ (Hours Unsupervised): $2, 2.5, 3.5, 4, 5.5$
• $y$ (Overall Grades): $99, 81, 73, 67, 63$

1. Compute the Means ($\bar{x}$ and $\bar{y}$):

$\bar{x} = \frac{\sum x_i}{n}, \quad \bar{y} = \frac{\sum y_i}{n}$

• $n = 5$
• $\bar{x} = \frac{2 + 2.5 + 3.5 + 4 + 5.5}{5} = 3.5$
• $\bar{y} = \frac{99 + 81 + 73 + 67 + 63}{5} = 76.6$

2. Compute $b_1$ (Slope):

$b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$ First, calculate $x_i - \bar{x}$ and $y_i - \bar{y}$, then multiply and square:

$x_i$	$y_i$	$x_i - \bar{x}$	$y_i - \bar{y}$	$(x_i - \bar{x})(y_i - \bar{y})$	$(x_i - \bar{x})^2$
2	99	$-1.5$	$22.4$	$-33.6$	$2.25$
2.5	81	$-1.0$	$4.4$	$-4.4$	$1.0$
3.5	73	$0.0$	$-3.6$	$0.0$	$0.0$
4	67	$0.5$	$-9.6$	$-4.8$	$0.25$
5.5	63	$2.0$	$-13.6$	$-27.2$	$4.0$

• $\sum (x_i - \bar{x})(y_i - \bar{y}) = -33.6 - 4.4 + 0.0 - 4.8 - 27.2 = -70.0$
• $\sum (x_i - \bar{x})^2 = 2.25 + 1.0 + 0.0 + 0.25 + 4.0 = 7.5$

$b_1 = \frac{-70.0}{7.5} = -9.333$

3. Compute $b_0$ (Intercept):

$b_0 = \bar{y} - b_1 \bar{x}$

• $\bar{y} = 76.6$, $\bar{x} = 3.5$, $b_1 = -9.333$ $b_0 = 76.6 - (-9.333)(3.5) = 76.6 + 32.666 = 109.266$

Final Answer:

The estimated intercept is: $b_0 = 109.266$

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Would you like further details or have any questions?

Related Questions:

1. How do you interpret the slope ($b_1$) in the context of this data?
2. What is the significance of the intercept ($b_0$) for these grades?
3. How can we determine if the correlation coefficient is statistically significant?
4. What are the limitations of using this regression line for predictions?
5. How would the calculations change if another data point was added?

Tip:

Always visualize your data with a scatterplot before performing regression to check for linearity!