Question

A 90 percent confidence interval for the slope of a regression line is determined to be (-0.181, 1.529). Which of the following statements must be true?
a. The correlation coefficient of the data is positive.
b. The sum of the residuals for the data based on the regression line is positive.
c. A scatterplot of the data would show a linear pattern.
d. The slope of the sample regression line is 1.348.
e. The slope of the sample regression line is 0.

Answer 1

Answer: a. The correlation coefficient of the data is positive.

Step-by-step explanation:

Estimated slope of sample regression line  = $\dfrac{Upper \ limit + lower\ limit}{2}$

Here , confidence interval :  (-0.181, 1.529)

Estimated slope of sample regression line  = $\dfrac{-0.181+1.529}{2}$

$=\dfrac{1.348}{2}\\\\=0.674\ \ \ \ [\text{ positive}]$

⇒Correlation coefficient(r) must be positive, So a. is true.

But, d. and e. are wrong(0.674 ≠ 0 or 1.348).

We cannot check residuals or its sum from confidence interval of slope of a regression line, so b is wrong.

We cannot say that scatterplot is linear as we cannot determine it from interval, so c. is wrong

So, the correct option : a. The correlation coefficient of the data is positive.