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# How to Create a Prediction Interval in R

A linear regression modelÂ can be useful for two things:

(1) Quantifying the relationship between one or more predictor variables and a response variable.

(2)Â Using the model to predict future values.

In regards toÂ (2), when we use a regression model to predict future values, we are often interested in predicting both anÂ exact valueÂ as well as anÂ intervalÂ that contains a range of likely values. This interval is known as aÂ prediction interval.

For example, suppose we fit a simple linear regression model usingÂ hours studiedÂ as a predictor variable andÂ exam scoreÂ as the response variable. Using this model, we might predict that a student who studies for 6 hours will receive an exam score of 91.

However, because there is uncertainty around this prediction, we might create a prediction interval that says there is a 95% chance that a student who studies for 6 hours will receive an exam score between 85 and 97. This range of values is known as a 95% prediction interval and itâ€™s often more useful to us than just knowing the exact predicted value.

## How to Create a Prediction Interval in R

To illustrate how to create a prediction interval in R, we will use the built-inÂ mtcarsÂ dataset, which contains information about characteristics of several different cars:

```#view first six rows of mtcars

#                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
#Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
#Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
#Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
#Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
#Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
#Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1
```

First, weâ€™ll fit a simple linear regression model usingÂ dispÂ as the predictor variable andÂ mpgÂ as the response variable.

```#fit simple linear regression model
model #view summary of fitted model
summary(model)

#Call:
#lm(formula = mpg ~ disp, data = mtcars)
#
#Residuals:
#    Min      1Q  Median      3Q     Max
#-4.8922 -2.2022 -0.9631  1.6272  7.2305
#
#Coefficients:
#             Estimate Std. Error t value Pr(>|t|)
#(Intercept) 29.599855   1.229720  24.070  ```

Then, weâ€™ll use the fitted regression model to predict the value of mpgÂ based onÂ three new values forÂ disp.Â

```#create data frame with three new values for disp
new_disp
#use the fitted model to predict the value for mpg based on the three new values
#for disp
predict(model, newdata = new_disp)

#       1        2        3
#23.41759 21.35683 19.29607
```

The way to interpret these values is as follows:

• For a new car with a dispÂ of 150, we predict that it will have aÂ mpgÂ of 23.41759.Â
• For a new car with a dispÂ of 200, we predict that it will have aÂ mpgÂ of 21.35683 .Â
• For a new car with a dispÂ of 250, we predict that it will have aÂ mpgÂ of 19.29607.Â

Next, weâ€™ll use the fitted regression model to make prediction intervals around these predicted values:

```#create prediction intervals around the predicted values
predict(model, newdata = new_disp, interval = "predict")

#       fit      lwr      upr
#1 23.41759 16.62968 30.20549
#2 21.35683 14.60704 28.10662
#3 19.29607 12.55021 26.04194
```

The way to interpret these values is as follows:

• The 95% prediction interval of theÂ mpgÂ for a car with aÂ dispÂ of 150 is between 16.62968 and 30.20549.
• The 95% prediction interval of theÂ mpgÂ for a car with aÂ dispÂ of 200 is between 14.60704Â and 28.10662.
• The 95% prediction interval of theÂ mpgÂ for a car with aÂ dispÂ of 250 is between 12.55021Â and 26.04194.

By default, R uses a 95% prediction interval. However, we can change this to whatever weâ€™d like using theÂ levelÂ command. For example, the following code illustrates how to create 99% prediction intervals:

```#create 99% prediction intervals around the predicted values
predict(model, newdata = new_disp, interval = "predict", level = 0.99)

#       fit      lwr      upr
#1 23.41759 14.27742 32.55775
#2 21.35683 12.26799 30.44567
#3 19.29607 10.21252 28.37963
```

Note that the 99% prediction intervals are wider than the 95% prediction intervals. This makes sense because the wider the interval, the higher the likelihood that it will contain the predicted value.

## How to Visualize a Prediction Interval in R

The following code illustrates how to create a chart with the following features:

• A scatterplot of the data points forÂ dispÂ andÂ mpg
• A blue line for the fitted regression line
• Gray confidence bands
• Red prediction bands
```#define dataset
data #create simple linear regression model
model #use model to create prediction intervals
predictions predict")

#create dataset that contains original data along with prediction intervals
library(ggplot2)

#create plot
ggplot(all_data, aes(x = disp, y = mpg)) + #define x and y axis variables