Calculate linear regression (1,1)(2,3)(3,2)(4,4)(5,5)

Linear regression, confidence intervals, expected value, and covariance.

Data

Answer

y = 0.9x + 0.3 (R²=0.81)

To find the line of best fit for 5 data points using least squares regression, we need the slope (m) and y-intercept (b) of y = mx + b.

Step 1 — List the data points

(1, 1), (2, 3), (3, 2), (4, 4), (5, 5)

Step 2 — Compute the required sums

n = 5

Σx = 15, Σy = 15

Σxy = 54, Σx² = 55

Step 3 — Calculate the slope (m)

Using the least squares formula:

m = (nΣxy − ΣxΣy) / (nΣx² − (Σx)²)

= (5 × 54 − 15 × 15) / (5 × 55 − 15²)

= 45 / 50

m = 0.9

Step 4 — Calculate the y-intercept (b)

b = (Σy − mΣx) / n

= (15 − 0.9 × 15) / 5

b = 0.3

Step 5 — Measure the fit

The correlation coefficient r = 0.9 measures how closely the data follows a linear pattern.

R² = 0.81

This means 81% of the variation in y is explained by x.

Interpretation: strong positive correlation.

y = 0.9x + 0.3 (R² = 0.81)

To compute linear regression (1,1)(2,3)(3,2)(4,4)(5,5), apply the statistical formula to the data points.

This is a statistical analysis — advanced statistics includes regression, hypothesis testing, and measures of association.

What is the answer to linear regression (1,1)(2,3)(3,2)(4,4)(5,5)?
The answer is y = 0.9x + 0.3 (R²=0.81).

What method is used?
Linear regression finds y = mx + b that best fits the data. R² measures how well the line fits.

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