Testing a sample mean

The sample mean is a random variable. This equation shows how we can use this random quantity to test a hypothesised value for a population mean.

For large enough sample sizes, the Central Limit Theorem tells us that the variability of the sample proportion is approximately Normal. If we have a value that is 2 standard deviations above the mean then we know that this, or a more extreme value, could only occur about 2.5% of the time by chance. Unfortunately we usually don't know the standard deviation of the sample mean because we don't know the population standard deviation. We can estimate this with the sample standard deviation, giving the standard error for the sample mean, but then we have to use the t distribution.

Now the mean of the sample mean is the population mean, for which we have a hypothesised value. So we can calculate the number of standard errors we are from the mean by subtracting the hypothesised value and dividing by the standard error.

Suppose someone claimed that female UQ students had an average height of 165 cm. To test this we take a sample of 5 female UQ students and measure their heights, recording 172, 157, 174, 174, and 165 cm. This gives a sample mean of 168.4 cm with standard error 3.30 cm. Now 168.4 cm is 1.03 standard errors from the hypothesised 165 cm. The probability of being 1.19 standard errors away in t(4) is 30%. This is quite likely, so our data is consistent with a mean of 165 cm.

We could also answer test this claim by finding a confidence interval for the mean and looking to see if the hypothesised value is in that range.