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 t-distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly.
- 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].