How can I calculate confidence interval for a mean in R not using confint

Question

I am trying to work on a problem where I try generate random exponential and uniform distributions and sample from them. Then I calculate the confidence interval of the linear model on them using confint(); however, I don't know how to get the correct confidence interval levels that I got from confint() using mean, sd and qt.

Here is what I have so far:

x <- rexp(30, rate=1); x
confint(lm(x~1))
summary(lm(x~1))$coefficients
mean(x)
sd(x)
x.std.error <- sd(x)/sqrt(30)

I'm also not sure, how to do this using the runif command so if I can get help with that it would be really helpful

Answer 1

The result of confint in this context is just the ordinary classical 95% confidence interval for a population mean . The interval is centered around the sample mean ( mean(x) ), and the margin of error is the standard error you found ( x.std.error ) with a multiplier that comes from the t-distribution ( qt(0.975, 29) ). (This gives the 97.5th percentile of the t-distribution with 29 degrees of freedom; in this context, "degrees of freedom" can be regarded as one less than the sample size.)

To recover the confidence interval provided by confint(lm(x~1)) , you can use:

mean(x) - qt(0.975, 29) * x.std.error
mean(x) + qt(0.975, 29) * x.std.error

or equivalently, and perhaps more intuitively:

mean(x) + qt(0.025, 29) * x.std.error   # qt(0.025, 29) = -qt(0.975, 29)
mean(x) + qt(0.975, 29) * x.std.error

I'm not quite sure what you mean when you say that you're not sure how to do this using runif , but presumably it's the same basic process as what you did, but replacing the first line with runif(30, 10, 15) for 30 variables uniformly distributed on the interval [10, 15] (as an example).