# 6.4: Using R to Find Confidence Intervals

- Page ID
- 220902

The confidence interval for a population’s mean, \(\mu\), given an experimental mean, \(\bar{x}\), for \(n\) samples is defined as

\[\mu = \bar{x} \pm \frac {z \sigma} {\sqrt{n}} \nonumber\]

if we know the population's standard deviation, \(\sigma\), and as

\[\mu = \bar{x} \pm \frac {t s} {\sqrt{n}} \nonumber\]

if we assume that the sample's standard deviation, \(s\), is a reasonable predictor of the population's standard deviation. To find values for \(z\) we use R's`qnorm`

`()`

function, which takes the form

`qnorm(p)`

where`p`

is the probability on one side of the normal distribution curve that a result is not included within the confidence interval. For a 95% confidence interval, \(p = 0.05/2 = 0.025\) because the total probability of 0.05 is equally divided between both sides of the normal distribution. To find \(t\) we use R's`qt`

`()`

function, which takes the form

`qt(p, df)`

where`p`

is defined as above and where`df`

is the degrees of freedom or \(n - 1\).

For example, if we have a mean of \(\bar{x} = 12\) for 10 samples with a known standard deviation of \(\sigma = 2\), then for the 95% confidence interval the value of \(z\) and the resulting confidence interval are

*# for a 95% confidence interval, alpha is 0.05 and the probability, p, on either end of the distribution is 0.025; *

*# the value of z is positive on one side of the normal distribution and negative on the other side; *

*# as we are interested in just the magnitude, not the sign, we use the abs() function to return the absolute value*

`z = qnorm(0.025)`

conf_int_pop = abs(z * 2/sqrt(10))

conf_int_pop

`[1] 1.23959`

Adding and subtracting this value from the mean defines the confidence interval, which, in this case is \(12 \pm 1.2\).

If we have a mean of \(\bar{x} = 12\) for 10 samples with an experimental standard deviation of \(s = 2\), then for the 95% confidence interval the value of \(t\) and the resulting confidence interval are

`t = qt(p = 0.025, 9)`

conf_int_samp = abs(t * 2/sqrt(10))

conf_int_samp

`[1] 1.430714`

Adding and subtracting this value from the mean defines the confidence interval, which, in this case is \(12 \pm 1.4\).