# 43 9.3 Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions. **Particular distributions are associated with hypothesis testing.**

Perform tests of a population mean using a **normal distribution** or a **Student’s t-distribution**.

(Remember, use a Student’s *t*-distribution when the population **standard deviation** is unknown and the distribution of the sample mean is approximately normal.)

We perform tests of a population proportion using a normal distribution (usually *n* is large or the sample size is large).

If you are testing a **single population mean**, the distribution for the test is for **means**:

[latex]\displaystyle\overline{{X}}[/latex] ~ [latex]{N}{\left(\mu_{{x}}\frac{{\sigma_{{x}}}}{\sqrt{{n}}}\right)}{\quad\text{or}\quad}{t}_{{df}}[/latex]

- The population parameter is
*μ*. - The estimated value (point estimate) for μ is [latex]\displaystyle\overline{{x}}[/latex], the sample mean.

If you are testing a **single population proportion**, the distribution for the test is for proportions or percentages:

[latex]\displaystyle{P'}[/latex] ~ [latex]{N}{\left({p,}\sqrt{{\frac{{{p}{q}}}{{n}}}}\right)}[/latex]

- The population parameter is
*p*. - The estimated value (point estimate) for
*p*is*p′*.

[latex]\displaystyle{p}\prime=\frac{{x}}{{n}}[/latex] where*x*is the number of successes and*n*is the sample size.

## Assumptions

When you perform a **hypothesis test of a single population mean *** μ* using a

**Student’s**(often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly.

*t*-distribution- Your data should be a
**simple random sample.** - Your data comes from a population that is approximately
**normally distributed**. - You use the sample
**standard deviation**to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t-test will work even if the population is not approximately normally distributed).

When you perform a **hypothesis test of a single population mean μ **using a normal distribution (often called a

*z*-test), the assumptions are:

- You take a simple random sample from the population.
- The population you are testing is normally distributed or your sample size is sufficiently large.
- You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a **hypothesis test of a single population proportion *** p*, you take a simple random sample from the population. You must meet the conditions for a

**binomial distribution**which are as follows:

- There are a certain number
*n*of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success*p*. The quantities*np*and*nq*must both be greater than five (*np*> 5 and*nq*> 5). - The shape of the binomial distribution needs to be similar to the shape of the normal distribution. The binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with
*μ*=*p*and [latex]\displaystyle\sigma=\sqrt{{\frac{{{p}{q}}}{{n}}}}[/latex]. Remember that*q*= 1 –*p*.

## Concept Review

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

When testing for a single population mean:

- A Student’s
*t*-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation. - The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.

When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean number of failures satisfy the conditions: *np* > 5 and *nq* > *n* where *n* is the sample size, *p* is the probability of a success, and *q* is the probability of a failure.

## Formula Review

If there is no given preconceived *α*, then use *α* = 0.05.

**Types of Hypothesis Tests:**

- Single population mean,
**known**population variance (or standard deviation):**Normal test**. - Single population mean,
**unknown**population variance (or standard deviation):**Student’s**.*t*-test - Single population proportion:
**Normal test**. - For a
**single population mean**, we may use a normal distribution with the following mean and standard deviation. Means: [latex]\displaystyle\mu=\mu_{{\overline{{x}}}}{\quad\text{and}\quad}\sigma_{{\overline{{x}}}}=\frac{{\sigma_{{x}}}}{\sqrt{{n}}}[/latex] - A
**single population proportion**, we may use a normal distribution with the following mean and standard deviation. Proportions: [latex]\displaystyle\mu={p}{\quad\text{and}\quad}\sigma=\sqrt{{\frac{{{p}{q}}}{{n}}}}[/latex].