How to compute a confidence interval for a population mean (in R)

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If we have a set of data that seems normally distributed, how can we compute a confidence interval for the mean? Assume we have some confidence level already chosen, such as $\alpha=0.05$.

We will use the $t$-distribution because we have not assumed that we know the population standard deviation, and we have not assumed anything about our sample size. If you know the population standard deviation or have a large sample size (typically at least 30), then you can use $z$-scores instead; see how to compute a confidence interval for a population mean using z-scores.

Solution

When applying this technique, you would have a series of data values for which you needed to compute a confidence interval for the mean. But in order to provide code that runs independently, we create some fake data below. When using this code, replace our fake data with your real data.

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alpha <- 0.05       # replace with your chosen alpha (here, a 95% confidence level)
data <- c( 435,542,435,4,54,43,5,43,543,5,432,43,36,7,876,65,5 ) # fake

# If you need the two values stored in variables for later use, do:
answer <- t.test( data, conf.level=1-alpha )
lower_bound <- answer$conf.int[1] upper_bound <- answer$conf.int[2]

# If you just need to see the results in a report, do this alone:
t.test( data, conf.level=1-alpha )

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One Sample t-test

data:  data
t = 3.1853, df = 16, p-value = 0.005753
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
70.29848 350.05446
sample estimates:
mean of x
210.1765


Note: The solution above assumes that the population is normally distributed, which is a common assumption in introductory statistics courses, but we have not verified that assumption here.