When an environmental company went out to measure the radon level at a house it found that the radon exceeded the standard level. The environmental company decided to measure the average amount of radon detected by randomly checking the house’s water supply twenty times over the next two months.
The graphical display of the data was approximately normal. The environmental company ran a test to see if there was evidence the average amount of radon in this house was significantly higher than it should be.
Why should the company run a $t$-test rather than a $z$-test?