www.john-weber.com  

Chapter 7: Inference for Distributions

Section 7.1: Inference for the Mean of a Population

In the previous chapter we assumed that we knew σ of the population. This is not realistic. But σ is important because it affects hypothesis testing and CI! Here are the assumptions of the z–test:

  1. data is from an SRS of size n from the population
  2. the population is N(μ, σ)
  3. the sampling distribution is N(μ, √σ/n)
Thus, the z–statistic is: .

What if σ is unknown? Then we estimate σ with the standard error (SE) of the sample mean: .

The t distributions

assumptions of the t distributions:

  1. data is from an SRS of size n from the population
  2. the population is N(μ, σ) but σ is unknown
Then the one–sample t–statistic is: , where s is the sample standard deviation.

There is a different t distribution for each sample size. So, to identify the t distribution we use the degrees of freedom (df). The df for a t distribution is n – 1. We represent the t distribution as t(n – 1).

The density curve of t(k) is given by: , where c is a constant, k is df. The t distribution looks like the standard normal, N(0, 1), curve except that the t distribution is wider. As k increases, the t(k) curve approaches the N(0, 1) curve. To find critical values for the t distribution see Table C of your text.

The t confidence intervals and test

NOTE: the one–sample t–statistic procedures are analogous to the procedures for the one–sample z–statistic!

A level C CI for μ is x-bar ± t*×SE. This interval is exact when the population distribution is normal, and is approximately correct for large n in other cases. Luckily, the TI83 can calculate the t CI!

t–test for a population mean:

  1. State the hypotheses:
  2. Calculate the P–value
  3. Make your conclusion

Recall: H0 and Ha always refer to the population and NOT to a particular outcome. It is often easier (and more appropriate) to state H0 and Ha before looking at the data.

Matched pairs t procedures

Matched pairs is an experimental design that collects before-and-after observations on the same subjects. Apply the one–sample t procedures to the observed differences. Luckily, the TI83 can perform matched pairs t procedures!

Robustness of t procedures

A statistical inference procedure is called robust if the probability calculations do not change very much when the assumptions of the procedure are violated.


Back to John Weber's MATH 1431 Page
Back to john-weber.com