Reading 11- LOS F (Part 2): : Q1 single, against, across, style

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1．A variable y is regressed against a single variable x across 28 observations. The value of the slope is 1.89, and the constant is 1.1. The mean value of x is 1.50, and the mean value of y is 3.94. The standard deviation of the x variable is 0.96, and the standard deviation of the y variable is 2.85. The regression sum of squares is 79.07, and the total sum of squares is 195.24. For an x value of 2.0, what is the 95% confidence interval for the y value?

A)   0.41 to 9.35.

B)   1.01 to 8.75.

C)   1.21 to 8.65.

D)   1.50 to 8.36.

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The correct answer was A)

First the standard error of the estimate must be calculated—it is equal to the square root of the mean squared error, which is equal to the residual sum of squares divided by the number of observations minus 2. The residual sum of squares is equal to the difference between the total sum of squares and the regression sum of squares, which is 195.24 − 79.07 = 116.17. The standard error of the estimate is equal to (116.17 / 26)^1/2 = 2.11. The standard deviation of the prediction is equal to the standard error of the estimate multiplied by {1 + 1/n + (X −X )²/[(n − 1)s_x²]}^1/2 = 2.11 × [1 + 1/28 + (2.0 − 1.5)²/(27 × 0.96²)]^1/2 = 2.17. The prediction value is 1.1 + (2.0 × 1.89) = 4.88. The t-value for 26 degrees of freedom is 2.056. The endpoints of the interval are 4.88 ± 2.056 × 2.17 = 0.41 and 9.35.