Choosing Distributions and Checking Assumptions for Hypothesis Tests

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When is a z-test used for a single population mean?

When the population standard deviation $\sigma$ is known. The test uses the normal distribution and the sample mean as the point estimate of the population mean.

When is a one-sample t-test used?

When testing a population mean and $\sigma$ is unknown. The sample standard deviation $s$ estimates $\sigma$, and a one-sample t-test typically has $n-1$ degrees of freedom.

What assumptions support a normal approximation for a proportion test?

The data satisfy binomial conditions: a fixed number of independent trials, two outcomes per trial, and a constant success probability. Expected successes and failures should each be sufficiently large; the stated check is $np>5$ and $nq>5$.

What is the standard error of the sample mean when $\sigma$ is known?

It is $\sigma/\sqrt{n}$. This measures the typical sampling variability of the sample mean.

What is the standard error of a sample proportion?

It is $\sqrt{pq/n}$, where $q=1-p$. For a hypothesis test, use the null-model standard error based on the hypothesized proportion.

Quiz questions

1. Which distribution is used to test a population mean when the population standard deviation is known?Answer: The normal distribution, as in a z-test.
2. Which sample statistic is the point estimate of a population mean?Answer: The sample mean, $\bar{x}$.
3. Why does a mean test with unknown population standard deviation use Student’s t-distribution?Answer: It accounts for added uncertainty from estimating $\sigma$ with the sample standard deviation.
4. For a proportion model with $p=0.2$, $q=0.8$, and $n=40$, do the stated checks $np>5$ and $nq>5$ hold?Answer: Yes; $np=8$ and $nq=32$, so both exceed 5.
5. A researcher tests one population mean, does not know $\sigma$, and has a simple random sample. Which test and estimate are appropriate?Answer: A t-test; estimate $\sigma$ using the sample standard deviation $s$.