The data from exercise 3 follow. xi 2 6 9 13 20 yi 7

The data from exercise 3 follow. xi 2 ...

Key Concepts

Simple Linear Regression

This is a statistical method used to model the relationship between a response variable and one predictor variable by fitting a linear equation. The model estimates parameters such as the slope and intercept, which define the line that best describes the trend in the data using the least squares method.

Confidence Interval for the Mean Response

This interval estimates the range within which the average response value is expected to fall at a specific predictor value. It accounts for the uncertainty in estimating the true linear relationship and is generally narrower because it reflects the variability in the mean response rather than the full spread of individual observations.

Prediction Interval for a New Observation

This interval provides a range that is likely to contain the value of a single new observation at a given predictor value. It incorporates both the uncertainty in estimating the mean response and the individual variability in the outcomes, making it wider than the confidence interval for the mean response.

Difference Between Confidence and Prediction Intervals

The key difference lies in their interpretation and calculation: confidence intervals estimate the uncertainty around the mean response for a given predictor, while prediction intervals account for the extra variability inherent in individual observations. Thus, prediction intervals are wider because they must capture both the uncertainty of the estimated regression line and the random error associated with individual responses.

Transcript

00:01 Since the data for this problem comes from exercise number three, we figured out already the expected regression equation.

00:08 So using that, i figured out what the y -hat values were.

00:11 I just plugged in each x value into the equation and came up with the y -hat values.

00:18 The expected y -values for each of these x values and came up with this.

00:22 So we need to figure out the confidence and prediction intervals when x equals 12 for these problems.

00:27 So what we need to do is first come up with the confidence.

00:30 Interval.

00:30 The confidence interval is the expected y value at x equals 12 plus the t test statistic of alpha over 2, where alpha is equal to 1 minus a confidence level of 0 .95, which is equal to 0 .05, so alpha is equal to 0 .05 times the expected standard deviation for our y hat star.

00:55 So first we need to find this value over here.

00:59 So this is equal.

01:01 To our standard deviation times the square root of 1 over n plus the difference between our given x value of 12 and our mean x squared divided by the square root of each of the individual x values and the mean x squared so first we need to find out what this s value is so s is equal to the square root of the mean square of the error.

01:32 The mean square of the error is equal to the sum of squares of the error divided by the number of elements minus two, and the sum of squares of the error is equal to the sum of each individual y value minus the expected y value at each corresponding point squared.

01:55 So what i mean by that is we would take 7 minus 9 .4 squared.

02:00 And 18 minus 13 squared and 9 minus 15 .7 squared.

02:06 And after we do, so i'll just write down an example of what we would do, so 7 minus 9 .4 squared.

02:19 And we would do that till the very end.

02:21 And ultimately we get a sum of squares of the error equaling 127 .3.

02:32 So with this 127 .3, we need to find out the number of elements that we have.

02:41 Actually, let me do this in red, 127 .3 divided by the number of elements minus 2.

02:49 And we have five elements, that is equal to 5.

02:53 Or we have five columns in this set.

02:56 So 5 minus 2 is 3, so 127 .3 divided by 3.

03:02 Which is equal to 42 .43 repeating.

03:10 And now we take the square root of 42 .43 repeating to get a standard deviation of 6 .5149.