Use the results of Problem 14 in Section 14.1 to answer the...

Use the results of Problem 14 in Section 14.1 to answer the following questions: (a) Predict the mean wind speed of all hurricanes whose atmospheric pressure is $950 \mathrm{mb}$. (b) Construct a $95 \%$ confidence interval for the mean wind speed found in part (a). (c) Predict the wind speed of a randomly selected hurricane whose atmospheric pressure is $950 \mathrm{mb}$. (d) Construct a $95 \%$ prediction interval for the wind speed found in part (c). (e) Explain why the predicted lengths in parts (a) and (c) are the same, yet the intervals constructed in parts (b) and (d) are different.

Use the results of Problem 14 in Section 14.1 to answer the following questions:
(a) Predict the mean wind speed of all hurricanes whose atmospheric pressure is $950 \mathrm{mb}$.
(b) Construct a $95 \%$ confidence interval for the mean wind speed found in part (a).
(c) Predict the wind speed of a randomly selected hurricane whose atmospheric pressure is $950 \mathrm{mb}$.
(d) Construct a $95 \%$ prediction interval for the wind speed found in part (c).
(e) Explain why the predicted lengths in parts (a) and (c) are the same, yet the intervals constructed in parts (b) and (d) are different.

Key Concepts

Difference Between Confidence and Prediction Intervals

This concept distinguishes the two types of intervals by clarifying that a confidence interval estimates the average outcome while a prediction interval reflects the expected range of individual outcomes. Although the point prediction (mean prediction) remains the same in both cases, the prediction interval is broader because it accounts for extra variability due to individual differences, whereas the confidence interval solely focuses on the uncertainty in estimating the regression line.

Prediction Interval for an Individual Response

A prediction interval provides a range within which a new individual observation is expected to fall, given a specific value of the independent variable. In addition to the uncertainty in the mean response estimation, the prediction interval incorporates the error or variability observed in individual data points. As a result, prediction intervals are usually wider than confidence intervals.

Simple Linear Regression

This concept involves modeling the relationship between a dependent variable and one independent variable using a linear equation. In the context of the problem, the regression model is used to predict wind speed based on atmospheric pressure. It includes estimating the slope and intercept from sample data to make predictions about the mean behavior of the response variable.

Confidence Interval for the Mean Response

A confidence interval for the mean response is an estimated range of values that is likely to include the true average of the dependent variable for a given value of the independent variable. This interval accounts for the uncertainty in estimating the regression parameters, and it is typically narrower than a prediction interval because it does not include the additional variability inherent in individual observations.

Transcript

00:01 So i've input the values into my calculator and we are ready to use and predict what the mean wind speed is for hurricanes whose atmospheric pressure is 950.

00:12 And so we'll plug into the model that 1245 .94 minus 1 .19938 times that 950.

00:25 And so when i do that, let me quickly get that on my calculator here, times 950.

00:34 I find that it would predict 106 .528.

00:39 And i'm just going to take that value and store it as alpha a in my calculator.

00:44 Now on part b, we're asked to determine a 95 % confidence interval for the mean wind speed for that.

00:52 And so we're going to take that 106 .528 and the number of data points we have, let me quickly count those values for you or a quick look at my calculation here.

01:09 Shows that we have a 20 values.

01:13 So our t value for 95 % confidence is going to have .025 in the upper tail and 19 degrees of freedom.

01:21 And that value is a 2 .093.

01:26 And then we need to multiply that by the standard deviation of the air.

01:32 And that standard air was, or the standard deviation of the response, the residuals is that 7 .91.

01:42 And then we have times the square root of 1 over 20 plus we're going to take the x value 950 minus the mean, which i have as 985 .25 quantity squared.

01:58 And then i've already calculated the sum of each of the x values minus the mean squared and get that to be 8 ,809 .75.

02:11 So those are my numbers.

02:13 And i'm going to quick get those values for you.

02:15 I'm going to pause the video and quick get those numbers for you and then i'll give you those answers...

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