"Confidence Intervals for y-Intercept and Slope
You ca...

Question

"Confidence Intervals for y-Intercept and Slope

You can construct confidence intervals for the y-intercept B and slope M of the regression line y = Mx + B for the population by using the inequalities below.

y-intercept B :

b - E < B < b + E

where

E = t_c s_e \sqrt{\frac{1}{n} + \frac{\overline{x}^2}{\sum x^2 - \frac{(\Sigma x)^2}{n}}}

slope M :

m - E < M < m + E

where

E = \frac{t_c s_e}{\sqrt{\sum x^2 - \frac{(\Sigma x)^2}{n}}}

The values of m and b are obtained from the sample data, and the critical value t_c is found using Table 5 in Appendix B with n - 2 degrees of freedom.

In Exercises 37 and 38, construct the indicated confidence intervals for B and M using the gross domestic products and carbon dioxide emissions data found in Example 2.

38. 99% confidence interval"

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A biologist analyzes the correlation between average rainfall, X in units of centimeters, and plant height Y in units of centimeters, or N equals 15 regions. The regression line has a sample slope M equals 0.85. With S subscript E is equal to 0.19. The sum of X is equal to 540. The sum of X squared is equal to 20,500. Calculate the 99% confidence interval for the population slope M. Use T subscript C is equal to 2.977 or N minus 2 is equal to 13 degrees of freedom. OK. So it appears for this particular prompt, we're ultimately trying to determine, or rather calculate, the 99% confidence interval for the population slope M, provided all the information that is given to us by the prom itself. So ultimately we're trying to determine what this interval is, and that's our final answer we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our possible answer for the interval might be, noting that they're all written in inside a parentheses and interval notation. So A is 0.62.1.08. B is 0.68.1.02. C is 0.73.0.97, and finally, D is 0.83.0.87. Awesome. OK. So, our first step that we need to take in order to solve this problem is we need to calculate, as we should recall, we need to calculate. The sum of X2 minus parentheses the sum of X to the power of 2 divided by n, so plugging in our known intervals into this formula, we will find that the sum of X 2 is equal to 20,000. 500 minus the sum of X to the power 2, which the sum of X is equal to 540, and once again we're using the values that are given to us by the problem itself. So we know that the sum of X is equal to 540, and then we need to take it to the power of 2 divided by our value for n, which is equal to 15. Fantastic. So when we plug in this expression into our calculator, we will find that it's simply equal to 1060. Fantastic. Our second step now is we need to compute the margin of error for this particular slope. So once again we're trying to determine the margin of error for the slope, so we'll call the margin of error E, but since we're specifically solving for the error for the slope, we'll call it E subscript M because once again it's the margin of error for the slope, and this will be equal to, as we should recall, the equation to solve for the margin of error will be equal to T subscript C multiplied by S subscript E. So we're going to take S subscript E, all divided by the square root of. The sum of X2 minus parenthesis, the sum of X 2 divided by n. Fantastic. And when we plug in our known variables, we will note that this is equal to 2.977 multiplied by 0.19. Once again, we're plugging in our known variables. So I've gone ahead and took the value for T subscript C, and I wrote it up above the division. But don't get confused. You could write it outside of the division bar if it makes more sense to you, but both ways are correct. So we're going to take our value for T subscript C, which is 2.977, multiplied by your value of S subscript E, which is 0.9, all divided by, and since we already determined. The value for the sum of X2 minus parenthesis, the sum of X 2 divided by n, we just need to take the square root of 1060. And when we do, when we take 2.977 multiplied by 0.19. divided by the square root of 1060, we will find that the margin of error for the slope is approximately equal to, because once again we're just plugging in that expression into our calculator, and when we do, we will find that it's approximately equal to 0.0174, and that's when we round to 4 decimal places. Fantastic. So now, moving right along here. Our third step now is we need to construct the confidence interval for the slope. So for our lower our lower bound, which I'll call LB for short, so focusing on our lower bound first, we need to take 0. 85 and we need to subtract 0.0174 and when we do, we will find that it is approximately equal to let me round to two decimal places, 0.83. And for our upper bound, similarly, we need to take 0.85, but instead of subtracting, we need to add 0.0174. And when we do, we will find that it's approximately equal to 0.87 when we round to two decimal places. So now when we write our final answer in interval notation, we will find that it's equal to parenthesis. So once again, Writing our final answer in interval notation, we will get 0 point. 83, 0.87 parenthesis because once again we're writing it in interval notation, and that's it. That's our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter D, which once again is parentheses 0.83.0.87. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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