# Practice 9 Calculating Confidence Intervals in R

## 9.1 Directions

In this practice exercise, you will calculate a confidence interval in R.

## 9.2 A closer look at the code

A confidence interval is an interval that contains the population parameter with probability \(1-\alpha\). A confidence interval takes on the form: \[\bar X \pm {t_{\alpha /2,N - 1}}{S_{\bar X}}\] where \(t_{\alpha /2,N - 1}\) is the value needed to generate an area of α/2 in each tail of a t-distribution with n-1 degrees of freedom and \({S_{\bar X}} = \frac{s}{{\sqrt N }}\) is the standard error of the mean.

### 9.2.1 Calculate a confidence interval

To calculate a confidence interval, we need the following steps

- Calculate the mean
- Calculate the standard error of the mean
- Find the t-score that corresponds to the confidence level
- Calculate the margin of error and construct the confidence interval

#### 9.2.1.1 Step 1: Calculate the mean

Use the `mean()`

command to calculate the average mpg.

```
data("mtcars")
mean(mtcars$mpg)
sample.mean <-print(sample.mean)
```

`## [1] 20.09062`

#### 9.2.1.2 Step 2: Calculate the standard error of the mean

The formula for the standard error of the mean is \({S_{\bar X}} = \frac{\sigma}{{\sqrt N }}\), and if we do not know the population standard deviation \({S_{\bar X}} = \frac{s}{{\sqrt N }}\).

The `sd()`

command can be used to find the standard deviation. The `length()`

command can be use to determine the sample size.

```
length(mtcars$mpg)
sample.n <- sd(mtcars$mpg)
sample.sd <- sample.sd/sqrt(sample.n)
sample.se <-print(sample.se)
```

`## [1] 1.065424`

#### 9.2.1.3 Step 3: Find the t-score that corresponds to the confidence level

We need to have \(\alpha/2\) probability in the lower and upper tails, we divide by two because there are two tails. The `qt()`

command will calculate the t-score, \(t_{\alpha /2,N - 1}\).

```
0.05
alpha = sample.n - 1
degrees.freedom = qt(p=alpha/2, df=degrees.freedom,lower.tail=F)
t.score =print(t.score)
```

`## [1] 2.039513`

#### 9.2.1.4 Step 4. Calculate the margin of error and construct the confidence interval

The margin of error is \({t_{\alpha /2,N - 1}}{S_{\bar X}}\)

` t.score * sample.se margin.error <-`

The confidence interval is the mean +/- margin of error

```
sample.mean - margin.error
lower.bound <- sample.mean + margin.error
upper.bound <-print(c(lower.bound,upper.bound))
```

`## [1] 17.91768 22.26357`

## 9.3 R code used in the VoiceThread

```
data("mtcars")
mean(mtcars$mpg)
sample.mean <-print(sample.mean)
length(mtcars$mpg)
sample.n <- sd(mtcars$mpg)
sample.sd <- sample.sd/sqrt(sample.n)
sample.se <-print(sample.se)
0.05
alpha = sample.n - 1
degrees.freedom = qt(p=alpha/2, df=degrees.freedom,lower.tail=F)
t.score =print(t.score)
t.score * sample.se
margin.error <- sample.mean - margin.error
lower.bound <- sample.mean + margin.error
upper.bound <-print(c(lower.bound,upper.bound))
```

## 9.4 A much easier way:

Let’s use linear regression as a short cut

```
# Calculate the mean and standard error
lm(mpg ~ 1, mtcars)
l.model <-
# Calculate the confidence interval
confint(l.model, level=0.95)
```

```
## 2.5 % 97.5 %
## (Intercept) 17.91768 22.26357
```

## 9.5 Now you try

Use R to complete the following activities (this is just for practice you do not need to turn anything in).

Using the mtcars data set, find a 95% confidence interval for the average horsepower, `hp`

.