76 THE SIMPLE LINEAR REGRESSION MODEL (x – x)² y - y (x – F)(y – F) y 6. 1 2 2 3 1 4 Ex, = Ey, = E(x, – x) = E(x, – x)² E(y, – 5) = E(x, – x)(y; – y) = (a) Complete the entries in the table. Put the sums in the last row. What are the sample means F and ỹ? (b) Calculate b, and b, using (2.7) and (2.8) and state their interpretation. (c) Compute E1x, £z1xyi. Using these numerical values, show that E(xi – x)° = Ex} – NF and E(x – x)(yi – 5) = Exy; – Nãy (d) Use the least squares estimates from part (b) to compute the fitted values of y, and complete the remainder of the table below. Put the sums in the last row. Yi 6. 2 3 3 1 4 Ex E = %3! %3D (e) On graph paper, plot the data points and sketch the fitted regression line ŷi = bị + b,xj. (f) On the sketch in part (e), locate the point of the means (X, §). Does your fitted line pass through that point? If not, go back to the drawing board, literally. (g) Show that for these numerical values y = b| + bx. (h) Show that for these numerical values y = y, where y = £ÿ¡/N.

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76
THE SIMPLE LINEAR REGRESSION MODEL
(x – x)²
y - y
(x – x)(y – F)
y
1
2
3
3
4
Ey
E(x; – x) =
E(x; – x)² =
Σύ-) -
E(x; – x)(y; – y) =
(a) Complete the entries in the table. Put the sums in the last row. What are the
sample means F and ỹ?
(b) Calculate bị and b2 using (2.7) and (2.8) and state their interpretation.
(c) Compute E1x}, £1xyi. Using these numerical values, show that
E(xi – x)° = Ex? – N and E(x; – x)(y; – 5) = E.xy: – Nay
(d) Use the least squares estimates from part (b) to compute the fitted values of y, and
complete the remainder of the table below. Put the sums in the last row.
6.
1
3
1
4
Ey =
Σ-
Ex;ê =
(e) On graph paper, plot the data points and sketch the fitted regression line
ŷi = bị + b,xj.
(f) On the sketch in part (e), locate the point of the means (x, y). Does your fitted line
pass through that point? If not, go back to the drawing board, literally.
(g) Show that for these numerical values y = b1 + bī.
(h) Show that for these numerical values ỹ = y, where y = £ÿ¡/N.
(i) Compute ở².
(j) Compute var(b2).
Transcribed Image Text:76 THE SIMPLE LINEAR REGRESSION MODEL (x – x)² y - y (x – x)(y – F) y 1 2 3 3 4 Ey E(x; – x) = E(x; – x)² = Σύ-) - E(x; – x)(y; – y) = (a) Complete the entries in the table. Put the sums in the last row. What are the sample means F and ỹ? (b) Calculate bị and b2 using (2.7) and (2.8) and state their interpretation. (c) Compute E1x}, £1xyi. Using these numerical values, show that E(xi – x)° = Ex? – N and E(x; – x)(y; – 5) = E.xy: – Nay (d) Use the least squares estimates from part (b) to compute the fitted values of y, and complete the remainder of the table below. Put the sums in the last row. 6. 1 3 1 4 Ey = Σ- Ex;ê = (e) On graph paper, plot the data points and sketch the fitted regression line ŷi = bị + b,xj. (f) On the sketch in part (e), locate the point of the means (x, y). Does your fitted line pass through that point? If not, go back to the drawing board, literally. (g) Show that for these numerical values y = b1 + bī. (h) Show that for these numerical values ỹ = y, where y = £ÿ¡/N. (i) Compute ở². (j) Compute var(b2).
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