Question

linear regression
below is data collected on how long statistics students study per week and the percentage of students who pass the course for the given amount of study time:

study time in hours per week (x):	0	2	6	9
% of students who pass statistics (y):	15	55	80	85

a.) would you say the data has a positive correlation or negative correlation?
∘ negative
∘ positive

b.) compute σx, σy, σx², σxy
σx =
σy =
σx² =
σxy =

c.) find the slope and the y-intercept of the best fit regression line. (round to 2 decimal places on slope and y-int)
y = mx + b
y = x +

d.) based on your answer from part c, what would you estimate the % completion rate is for students who study for 5 hours per week?
round to nearest %

Explanation:

Part a)

As study time (\(x\)) increases (from 0 to 9 hours), the percentage of students who pass (\(y\)) also increases (from 15 to 85). This indicates a positive correlation.

Step1: Calculate \(\sum x\)

Sum the \(x\)-values: \(0 + 2 + 6 + 9 = 17\)

Step2: Calculate \(\sum y\)

Sum the \(y\)-values: \(15 + 55 + 80 + 85 = 235\)

Step3: Calculate \(\sum x^2\)

Square each \(x\)-value and sum: \(0^2 + 2^2 + 6^2 + 9^2 = 0 + 4 + 36 + 81 = 121\)

Step4: Calculate \(\sum xy\)

Multiply each \(x\) and \(y\) and sum: \((0\times15)+(2\times55)+(6\times80)+(9\times85)=0 + 110 + 480 + 765 = 1355\)

Step1: Recall formulas for slope (\(m\)) and y - intercept (\(b\))

The formula for the slope \(m\) of the regression line is \(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\), and the formula for the y - intercept \(b\) is \(b = \frac{\sum y - m\sum x}{n}\), where \(n = 4\) (number of data points).

Step2: Calculate the slope \(m\)

Substitute \(n = 4\), \(\sum x=17\), \(\sum y = 235\), \(\sum xy=1355\), \(\sum x^{2}=121\) into the slope formula:

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Step3: Calculate the y - intercept \(b\)

Substitute \(m\approx7.31\), \(\sum x = 17\), \(\sum y=235\), \(n = 4\) into the y - intercept formula:

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Answer:

Positive

Part b)