MS Module 20 Standardized residuals ratio two points practice exam questions

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A regression model has the independent variable X values {1, 2, …, 7}.

●    At the point x = 3, the residual is 0.16 and the standardized residual is 0.11
●    The residual at the point x = 2 is -0.25


Question 20.1:

What is , the average X value?

Answer 20.1: (1 + 7) / 2 = 4


Question 20.2: Sxx

What is Sxx, the sum of squared residuals for the X values?

Answer 20.2: (1 – 4)2 + (2 – 4)2 + (3 – 4)2 + (4 – 4)2 + (5 – 4)2 + (6 – 4)2 + (7 – 4)2 = 28


Question 20.3: Ratio of residual to standardized residual

What is the ratio of the residual to the standardized residual at the point x = 3?

Answer 20.3: 0.16 / 0.11 = 1.455


Question 20.4: Standardized residual factor

At the point x = 3, what is the value of [(1 – 1/n – (xi – )2 / Sxx)]½ ?

Answer 20.4: (1 – 1/7 – (3 – 4)2 / 28)0.5 = 0.906


Question 20.5: Least squares estimate for σ

What is s, the least squares estimate for σ?

Answer 20.5: 1.455 / 0.906 = 1.606

(the standardized residual = the residual / ( σ × [(1 - 1/n - (xi - )2 / Sxx)]½ ) )


Question 20.6: Standardized residual

What is the standardized residual at the point x = 2?

Answer 20.6: -0.25 / (1.606 × (1 – 1 / 7 – (2 – 4)2 / 28)0.5 ) = -0.184

(the standardized residual = the residual / ( σ × [(1 - 1/n - (xi - )2 / Sxx)]½ ) )