Say we have a population whose mean $\mu$ is known. We take a sample $x_1,\ldots,x_n$ and compute its mean, $\bar x$. We then ask whether this sample is significantly different from the population at large, that is, is $\mu=\bar x$?
- How to compute a confidence interval for a population mean
- How to do a two-sided hypothesis test for two sample means
- How to do a one-sided hypothesis test for two sample means
- How to do a hypothesis test for a mean difference (matched pairs)
- How to do a hypothesis test for a population proportion
This is a two-sided test with the null hypothesis $H_0:\mu=\bar x$. We choose a value $0\leq\alpha\leq1$ as the probability of a Type I error (false positive, finding we should reject $H_0$ when it’s actually true).
1 2 3 4 5 6 7 8 # Replace these first three lines with the values from your situation. alpha <- 0.05 pop.mean <- 10 sample <- c( 9, 12, 14, 8, 13 ) # Run a one-sample t-test and print out alpha, the p value, # and whether the comparison says to reject the null hypothesis. t.test( sample, mu=pop.mean, conf.level=1-alpha )
1 2 3 4 5 6 7 8 9 10 One Sample t-test data: sample t = 1.0366, df = 4, p-value = 0.3585 alternative hypothesis: true mean is not equal to 10 95 percent confidence interval: 7.986032 14.413968 sample estimates: mean of x 11.2
Although we can deduce the answer to our question from the above output, by comparing the $p$ value with $\alpha$ manually, we can also ask R to do it.
1 2 3 # Is there enough evidence to reject the null hypothesis? result <- t.test( sample, mu=pop.mean, conf.level=1-alpha ) result$p.value < alpha
1  FALSE
In this case, the sample does not give us enough information to reject the null hypothesis. We would continue to assume that the sample is like the population, $\mu=\bar x$.
Content last modified on 24 July 2023.
Contributed by Nathan Carter (firstname.lastname@example.org)