U6: Linear Regression

General Notes

● Equation: ŷ = a + bx

○ b = Slope = Δy/Δx

● Line of Best Fit: Minimal residual (vertical distance between line and points)

● r: Shows correlation (not causation)

○ Sign of r = Sign of b (Slope)

○ The closer r is to ±1, the stronger r is

● r2: Correlation of determination (convert to a percentage)

● LSLR: Least squares regression line

● Extrapolation: Extending the LSLR to values outside the range of data used to generate it

Interpretations

● Slope (b)

○ For each [x], the [y] increases/decreases on average by [b]

● Correlation Coefficient (r)

○ There is a [strong/weak], [positive/negative], and [linear/non-linear] relationship between [x] and [y]

○ As [x] increases, it appears that [y] [increases/decreases].

● Correlation of determination (r2)

○ r2% variation in [y] can be explained by a linear relationship with [x]

● Put all interpretations in the context of the problem!

Residuals

● y - ŷ

● If the graph of the residuals shows a pattern, a non-linear model should be used

○ “Since the residual graph [does/does not] show a pattern, a [linear/nonlinear] model should be used”

● Positive residual = Underestimate; Negative residual = Overestimate

Table

● Line of best fit: ŷ = 1 + 5x

● 6 is the standard deviation of the slope (SEb)

● 7 is the t-score of the slope

● 8 is the p-value of 7

● 9 is the standard deviation of the residuals

Confidence Intervals

Step 1: Conditions

○ “The scatterplot shows a somewhat linear trend”

○ “A residual plot shows no pattern”

○ “All points are independent and random”

○ If the problem states that the conditions are satisfied, you can write: “The stem of the problem states the conditions are satisfied.”

Step 2: Formula

○ List the formula, substitution, unrounded answer, and degrees of freedom

○ Formula: b ± t * (SEb)

○ d.f. = n - 2

Step 3: Interpretation

○ “Based on these samples, we are _% confident that for every [x], [y] will [increase/decrease] between [lower value] and [upper value]”

Hypothesis Testing

Step 1: Hypothesis

○ H0: β = 0 (There is no linear relationship between [x] and [y])

○ Ha: β ≠ 0 (There is linear relationship between [x] and [y])

Step 2: Conditions

○ “The scatterplot shows a somewhat linear trend”

○ “A residual plot shows no pattern”

○ “All points are independent and random”

○ If the problem states that the conditions are satisfied, you can write: “The stem of the problem states the conditions are satisfied.”

Step 3: Formula

○ List the formula, substitution, unrounded answer, and degrees of freedom

○ Formula: t = (b - β)/SEb

○ d.f. = n - 2

Step 4: P Value

○ 2P(t > or < _) = _

Step 5: Conclusion

○ “Assuming H0 is true, since the p value [p value] is [greater/less] than α = _, we [fail to reject/reject] H0”

○ “We [have/do not have] sufficient evidence to suggest Ha, that there is a linear relationship between [x] and [y]”