Introductory Statistics 2e

# 9.3Probability Distribution Needed for Hypothesis Testing

Introductory Statistics 2e9.3 Probability Distribution Needed for Hypothesis Testing

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

The following table summarizes various hypothesis tests and corresponding probability distributions that will be used to conduct the test (based on the assumptions shown below):

Type of Hypothesis Test Population Parameter Estimated value (point estimate) Probability Distribution Used
Hypothesis test for the mean, when the population standard deviation is known Population mean $μμ$ Sample mean $x¯x¯$ Normal distribution, $X¯~N(μX,σXn)X¯~N(μX,σXn)$
Hypothesis test for the mean, when the population standard deviation is unknown and the distribution of the sample mean is approximately normal Population mean $μμ$ Sample mean $x¯x¯$ Student’s t-distribution, $tdftdf$
Hypothesis test for proportions Population proportion $pp$ Sample proportion $p'p'$ Normal distribution, $P'~N(p,p·qn)P'~N(p,p·qn)$
Table 9.3

## Assumptions

When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z-test), 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 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 that 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 proportion p, you take a simple random sample from the population. You must meet the conditions for a binomial distribution: 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 shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five ( and ). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with and $σ=pqnσ=pqn$. Remember that $q=1-pqq=1-pq$.