8.4 Distributions Required for a Hypothesis Test

Earlier in the course, we discussed sampling distributions:  the sampling distribution of the sample mean and the sampling distribution of the sample proportion.  These distributions play a role in hypothesis testing.

If the hypothesis test is on a population mean, we use the distribution of the sample means in the hypothesis test.  As we learned previously, the distribution of the sample means follows a normal distribution if the population the sample is taken from is normal or if the sample size is large enough ([latex]n \geq 30[/latex]).  For a hypothesis test on a population mean we use a normal distribution when the population standard deviation is known or a [latex]t[/latex]-distribution when the population standard deviation is unknown.

If the hypothesis test is on a population proportion, we use the distribution of the sample proportions in the hypothesis test.  As we learned previously, the distribution of the sample proportions follows a normal distribution if [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex] or a binomial distribution if one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex].  For a hypothesis test on a population proportion we use either a normal distribution or a binomial distribution, depending on which of the above conditions is met.

Assumptions

When we perform a hypothesis test of a single population mean[latex]\mu[/latex] and the population standard deviation is known, we take a simple random sample from the population.  We use a normal distribution, assuming the population is normal or the sample size is large enough ([latex]n \geq 30[/latex]).  The [latex]z[/latex]-score we need is the [latex]z[/latex]-score from the distribution of the sample means:  [latex]\displaystyle{z=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}}[/latex].

When we perform a hypothesis test of a single population mean[latex]\mu[/latex] and the population standard deviation is unknown, we take a simple random sample from the population.  We use a [latex]t[/latex]-distribution, assuming the population is normal or the sample size is large enough ([latex]n \geq 30[/latex]).  We use the sample standard deviation to approximate the population standard deviation.  The [latex]t[/latex]-score we need is:  [latex]\displaystyle{t=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}}[/latex].

When we perform a hypothesis test of a single population proportion [latex]p[/latex], we take a simple random sample from the population.  We use a normal distribution when [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex].  In this case, the [latex]z[/latex]-score we need is the [latex]z[/latex]-score from the distribution of the sample proportions:  [latex]\displaystyle{z=\sqrt{\frac{p \times (1-p)}{n}}}[/latex].  Otherwise, we use a binomial distribution when one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex].

Concept Review

In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.

Testing a population mean:

• Population standard deviation is known:  use a normal distribution, assuming the population is normal or [latex]n \geq 30[/latex].
• Population standard deviation is unknown:  use a [latex]t[/latex]-distribution, assuming the population is normal or [latex]n \geq 30[/latex].

Testing a population proportion:

• Use a normal distribution when [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex].
• Use a binomial distribution when at least one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex].

Attribution

“9.3 Distribution Needed for Hypothesis Testing“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.