A linear regression model predicts weekly revenue from ad spending. You find the prediction interval for exactly $200\$200$200 in ad spending is ($520,$610)\text{(\$}520,\$\text{610)}($520,$610). Choose the answer that best describes what this interval means.

We are 95 % 95\% 95% confident that a single weekly revenue value with $ 200 \$200 $200 in ad spending will fall between $ 520 \$520 $520 and $ 610 \$610 $610 .

A linear regression model predicts weekly revenue from ad spending. You find the prediction interval for exactly $ 200 \$200 $200 in ad spending is ($ 520 , $ 610) \text{(\$}520,\$\text{610)} ($ 520 , $ 610) . Choose the answer that best describes what this interval means.

Back, everybody. So let's take a look at this practice problem. It is a little bit different than the types of problems you'll typically see in a book. Those tend to be more calculation based, but it's a really good idea to look at these conceptual problems to get a good feel for what the stuff actually means in real words. All right? So let's get to get to get started here. We've got a linear regression model that predicts weekly revenue from ad spending. We don't know what the data actually is. All we know is that there there's a relationship between revenue and ad spending. Now we find the prediction interval for exactly $200 in ad spending, Remember, this is your x variable to be basically this prediction interval over here. So in other words, this is our prediction interval. We have a lower bound of $520 and an upper bound of 610. Now we have four answer choices over here, and we're going to choose the answer that best describes what this interval actually means in real words. Alright? So let's take a look. We're just gonna go one by one. So in choice a, we're told that the model generates at least $520 in revenue. Is this the best answer choice? Well, just remember what a prediction interval actually means on a graph. So I'm gonna just draw a little graph like this. Let's say I had a regression line that goes kinda goes up like this, and you were to sort of predict what a y value is going to be. So remember that that prediction isn't always with 100% confidence. This is kind of just our best guess. In other words, this is going to be a y zero hat, and there's always some margin of error that's associated with this. So the upper bound for this is y hats plus e, and then the lower bound of this is y hats minus e. And basically, what we're assuming over here is we construct these intervals, with some kind of a confidence level, either 95% or 90%, and that's gonna adjust, basically, how wide that prediction interval is. That's really what prediction intervals are about. Now, what this answer choice says is that the model generates at least $520 in revenue, which means that it could actually be more, and there's no really upper bound in that number. Now that's definitely not correct because, remember, with a prediction value interval, you always get a lower bound of your estimate and an upper bound of your estimate. This is at least 520 with no upper bound, so it definitely can't be the right answer. Alright? So answer choice a is out. Alright. So let's take a look at answer choice b. Now this says the average revenue for $200 in ad spending is exactly $565 Now where does that number come from? Well, basically, what happens is if you take the halfway or the middle number between these lower and upper bounds, five twenty and six ten, you're gonna get $565 Now what this means here is that's basically just the middle of your prediction interval, but that has nothing to do with the average revenue for this specific value. This is just what we predict using the regression line. It has nothing to do with what the actual average is based on the real data. So this is wrong as well. So really, the answer choice is going to come down to either C or D, and they're slightly different. So let's take a look. So this answer choice c, this says we're 95 confidence that a single weekly revenue value, and we're just gonna assume here this is a prediction interval of 95%, with $200 in ad spending will fall between 520 and $610 Right? So So in other words, we have a range here, $5.20 and $6.10. We're 95% confident that a single weekly revenue value of $200 is going to result in that interval. That seems pretty plausible. So let's take a look at the next answer choice to see which one might be more correct. The second one here says or sorry. This fourth one here says that we're 95% confident that the mean revenue from $200 in ad spending is between $5.20 and $6.10. So in other words, all the numbers are the same, and the only real difference between these is this phrase over here, single weekly revenue value and mean revenue of $200 And basically what's happening here is where the prediction interval, remember that a prediction interval is always of a single y value. It's a singular y value that you're using to basically establish a prediction interval based on your regression line. So the answer choice that is more correct is actually going to be answer choice C. So it's this one over here. Because the prediction interval always is about a single weekly revenue value, It doesn't tell you what the mean revenue is for that specific value. That's actually a slightly different problem that we'll cover in a later video. All right, so it's not answer choice D and answer choice C is correct. So let me know if you have any questions. Thanks for watching.

A linear regression model predicts weekly revenue from ad spending. You find the prediction interval for exactly $ 200 \$200 $200 in ad spending is ($ 520 , $ 610) \text{(\$}520,\$\text{610)} ($ 520 , $ 610) . Choose the answer that best describes what this interval means.

We are 95 % 95\% 95% confident that a single weekly revenue value with $ 200 \$200 $200 in ad spending will fall between $ 520 \$520 $520 and $ 610 \$610 $610 .

The model will generate at least $ 520 \$520 $520 in revenue.

The average revenue for $ 200 \$200 $200 in ad spending is exactly $ 565 \$565 $565 .

We are 95 % 95\% 95% confident the mean revenue from $ 200 \$200 $200 in ad spending is between $ 520 \$520 $520 and $ 610 \$610 $610 .

12. Regression

Prediction Intervals

Statistics for Business

12. Regression

Prediction Intervals

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Multiple Choice

A linear regression model predicts weekly revenue from ad spending. You find the prediction interval for exactly $\$200$ in ad spending is $\text{(\$}520,\$\text{610)}$ . Choose the answer that best describes what this interval means.

The model will generate at least $\$520$ in revenue.

The average revenue for $\$200$ in ad spending is exactly $\$565$ .

We are $95\%$ confident that a single weekly revenue value with $\$200$ in ad spending will fall between $\$520$ and $\$610$ .

We are $95\%$ confident the mean revenue from $\$200$ in ad spending is between $\$520$ and $\$610$ .

Verified step by step guidance

Understand the context: The problem involves interpreting a prediction interval in a linear regression model. A prediction interval provides a range where we expect a single observation (in this case, weekly revenue) to fall, given a specific value of the independent variable (ad spending).

Clarify the difference between prediction intervals and confidence intervals: A prediction interval is used to estimate the range for a single observation, while a confidence interval estimates the range for the mean of the dependent variable. This distinction is crucial for selecting the correct interpretation.

Analyze the given interval: The prediction interval for $200 in ad spending is ($520, $610). This means that the model predicts a single weekly revenue value to fall within this range with a certain level of confidence (typically 95%).

Evaluate the provided options: The first option ('The model will generate at least $520 in revenue') is incorrect because a prediction interval does not guarantee a minimum value; it only provides a range with a certain confidence level. The second option ('The average revenue for $200 in ad spending is exactly $565') is also incorrect because the interval does not describe the mean revenue. The fourth option ('We are 95% confident the mean revenue from $200 in ad spending is between $520 and $610') is incorrect because the interval is a prediction interval, not a confidence interval for the mean.

Select the correct answer: The third option ('We are 95% confident that a single weekly revenue value with $200 in ad spending will fall between $520 and $610') is correct because it accurately describes the purpose of a prediction interval in this context.

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