As the carbon content in steel increases, its ductility tends to decrease. A res

Question

As the carbon content in steel increases, its ductility tends to decrease. A researcher at a steel company measured content and ductility for a sample of 15 types of steel. Based on obtained the following regression results. Which of the following statements is not true? The regression equation is Ductility = 7.67 - 3.30 Carbon Content A. The slope is significantly different from zero at alpha = 0.05. B. The relationship between carbon content and ductility of steel is positive at alpha = 0.06 C. For one unit increase in carbon content, one can expect a 3.30 unit decrease in ductility. D. There is 5 6 of the variance in ductility that is not explained by this model.

Explanation / Answer

Solution

Let Y = ductility and X = carbon content.

The regression model is: y = 0 + 1x + .

The least square estimates of 0 and 1 are given in the question as:

0cap = 7.671 and 1cap = - 3.296.

Part (A) ANSWER: Statement is true.

Explanation: The given regression results output shows that the p-value for carbon content is 0.010 which is less than the given of 0.05. This implies that the null hypothesis that the slope is zero is rejected. This in turn implies that the slope is significantly different from zero.      

Part (B) ANSWER: Statement is NOT true.

Explanation: Part (A) shows that the slope, which represents the relationship between carbon content and ductility, is significantly different from zero. Further, the output shows the slope is negative (- 3.296). => the relationship between carbon content and ductility is negative (NOT positive) at the given of 0.05.

Part (C) ANSWER: Statement is true.

Explanation: The estimate 1cap = - 3.296 => for every increase of 1% in carbon content the ductility would come down (indicated by the negative sign) by 3.3 units.

Part (d) ANSWER: Statement is true.

Explanation: The given regression results output shows that the R-square = 41%. => 41% of variation in y (i.e., ductility) is explained by x (carbon content). This in turn implies that 59% of variation in y (i.e., ductility) is NOT explained by x (carbon content).

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