### Answer

A $95 \%$ confidence interval for the slope of a regression line is calculated to be $(-0.783,0.457) .$ Which of the following mustbe true?(A) The slope of the regression line is o.(B) The slope of the regression line is -0.326 .(C) A scatterplot of the data would show a linear pattern.(D) A residual plot would show no pattern.(E) The correlation is negative.

You are watching: Which of the following is true of the slope of the least-squares regression line?

### Video Transcript

All right. So in question for we are trying to figure out some information about a relationship between two quantitative variables, given only the 95% confidence interval for the slope of the least squares regression line. And so that interval is given here. And this is a question where reading the answer choices and reading all of them is really important. Now the first couple of answer choices are talking about what the slope of the least squares regression line would have been and I can actually figure that out using the interval that's given. So the midpoint of the interval that's given would have been the slope of the least squares regression line for the sample data. So I'm going to find the midpoint by essentially just averaging the two numbers together. So I would add up the end points And divide by two. And so this is going to give me negative 0.163. And so this is actually the slope of the least squares regression line from the sample. So I'm just identifying that as beef. So now I know that both answer choices a and b. Can be eliminated since the slope of the least squares regression line I now know was negative 0.163. Now. Andrew choice C says that the scatter plot shows a linear pattern and that's one of the conditions that has to be met in order to make this confidence interval in the first place. So I can eliminate C. And for the same reasoning, I can also eliminate answer choice D. Because the residual plot would have to show no pattern again in order um to meet a condition that would allow me to make this confidence interval. So then finally looking at answer choice E. Which is in fact the answer that's correct there at the correlation would be negative because if the slope is negative, that means that you would see in that scatter plot something kind of like this, a negative slope. And so that would mean that the correlation is also negative.

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So the correlation value our would be some negative number since the slope is negative. So that's the answer choice E.