# 9.3 Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions.

Perform tests of a population mean using a

(Remember, use a Student's

We perform tests of a population proportion using a normal distribution (usually

If you are testing a

If you are testing a

When you perform a

When testing for a single population mean:

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:

**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**:$\displaystyle\overline{{X}}$

~ ${N}{\left(\mu_{{x}}\frac{{\sigma_{{x}}}}{\sqrt{{n}}}\right)}{\quad\text{or}\quad}{t}_{{df}}$

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

If you are testing a

**single population proportion**, the distribution for the test is for proportions or percentages:$\displaystyle{P'}$

~ ${N}{\left({p,}\sqrt{{\frac{{{p}{q}}}{{n}}}}\right)}$

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

$\displaystyle{p}\prime=\frac{{x}}{{n}}$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***using a***μ****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**using a normal distribution (often called a*μ**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$\displaystyle\sigma=\sqrt{{\frac{{{p}{q}}}{{n}}}}$. 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:$\displaystyle\mu=\mu_{{\overline{{x}}}}{\quad\text{and}\quad}\sigma_{{\overline{{x}}}}=\frac{{\sigma_{{x}}}}{\sqrt{{n}}}$ - A
**single population proportion**, we may use a normal distribution with the following mean and standard deviation. Proportions:$\displaystyle\mu={p}{\quad\text{and}\quad}\sigma=\sqrt{{\frac{{{p}{q}}}{{n}}}}$.