Question

In each part, we have identified a hypothesis-testing procedure for one population mean. State the assumptions required and the test statistic used in each case.
a. one-mean$t-$test b.one-mean$z-$test

Short answer

$t-$test procedure can be used only on small sample size, whereas $z-$test procedure can be used on small, moderate and large sample sizes.

Step-by-step solution

Step 1. Given Information

Test statistic used in
a. one-mean $t-$test
b. one-mean $z-$test

Step 2. Conditions to use a.one-mean t-test

The procedure is for:
Small sample size

• Samples are randomly selected.
• Population follows normal distribution or the sample size is larger.
• The standard deviation is unknown.

The test statistic for one-mean $t-$test is given below:

$t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}$

Step 3. Conditions to use b. one-mean z-test

The procedure is for:
Small sample size:
If the sample size is say less than $15$, the$z-$test procedure is used when the variable is normally distributed or very close to being normally distributed.
Moderate sample size:
If the sample size lies between $15-30$, the$z-$test procedure is used when the variable is far from being normally distributed or there is no outlier in the data.
Large sample size:
If the sample size is say greater than $30$, the$z-$test procedure is used without any restriction.
The test statistic for one-mean$z-$test is given below:
$z=\frac{\overline{x}-{\mu }_0}{\frac{\sigma }{\sqrt{n}}}$