4.4 Setting Up a Test for a Population Mean or Population Mean Difference

Identify an appropriate testing method and parameter for a population mean or population mean difference with unknown σ.

• The appropriate test for a population mean with unknown population standard deviation σ is a one-sample t-test for a population mean.
• For a matched pairs design with two dependent samples, the appropriate analysis calculates differences between pairs of values to produce one sample of differences. The hypothesis testing procedure for the matched pairs design is a one-sample t-test for the population mean difference.
• The parameter for a hypothesis test for a population mean and population mean difference should reference the population parameter, the response variable, and the population in context.

Identify the null and alternative hypotheses for a population mean or population mean difference with unknown σ.

• The null hypothesis for a one-sample t-test for a population mean is H:00 ,μμ= in which µ0 is the null hypothesized value for the population mean. A one-sided alternative hypothesis for a one-sample t-test for a population mean is either H:a μμ< 0 or H:a μμ> 0. A two-sided alternative hypothesis is H:a μμ≠ 0.
• The null hypothesis for a population mean difference is H:0 0μd = . A one-sided alternative hypothesis for a population mean difference is either H:a μd < 0 or H:a μd > 0. A two-sided alternative hypothesis is H:a μd ≠ 0.

Justify the appropriateness of a hypothesis test for a population mean or population mean difference by verifying conditions.

• A one-sample t-test for a population mean or a population mean difference requires that three conditions be met:
  • i. The randomization condition—the data should be collected using a random sample or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, the population size must be at least 10 times larger than the sample size (%nN≤10 ), where N is the size of the population and n is the sample size.
  • iii. The sample data condition—it is indicated the population distribution is approximately normal, or n ≥30, or if n <30, the sample data distribution should be free from strong skewness and outliers. For matched pairs, the number of differences should be greater than or equal to 30. If the number of differences is less than 30, the sample of differences should be free from strong skewness and outliers. 129 Inference for Quantitative Data: Means UNIT 4