Chapter 6: Introduction to Formal Statistical Inference November 19, 2008

6.2 Large Sample Tests for a Mean While a confidence interval is an inferential technique that attempts to estimate a population parameter, a test of significance is an inferential technique used to determine the validity of some claim concerning a population based on sample data The reasoning behind tests of significance is based on what would happen over repeated sampling

The Basics 1. Some claim is made concerning the population of interest. 2. As a means of determining the validity of the claim, a sample is collected and the appropriate statistics and sampling distributions are obtained. 3. Assuming that the claim is true, one will determine how likely the obtained sample results are. 4. If one concludes that the obtained sample data are not likely to be obtained given the claim is true, then one will conclude the claim is false.

The Hypotheses Null Hypothesis: statement of the form Parameter = # Forms the basis of investigation in a significance test Usually formed to embody a status quo/ “pre- data” view of the parameter Denoted as Ho (H naught)

The Hypotheses Alternative Hypothesis is a statement that stands in opposition to the null hypothesis. Specifies what forms of departure from the null hypothesis are of concern (>, < or ≠) Denoted as Ha

The Hypotheses for a Test of Mean Three possible pairs of hypotheses: Ho: µ = # Ho: µ = # Ho: µ = # Ha: µ > # Ha: µ < # Ha: µ ≠ #

Test Statistic The particular form of numerical data summarization used in a significance test. The formula typically involves the number appearing in the null hypothesis. The reference distribution for the test statistic is the probability distribution describing the test statistic, provided the null hypothesis is in fact true.

Test Statistic for Large Sample Test where σ is Known

Step 3: p value The observed level of significance (p-value) is the probability that the reference distribution assigns to the set of possible values of the test statistic that are at least as extreme as the one actually observed In terms of casting doubt on the null hypothesis, small p-values are evidence against Ho

5 Step Format for Hypothesis Tests 1. State the null hypothesis 2. State the alternative hypothesis 3. State the test criteria (test statistic and reference distribution) 4. Show the sample-based calculations. 5. Report a p-value and state its implications in the context of the problem

Example: Baby Food Suppose the declared label weight of the baby food is 135 grams and process engineers have set a target mean net fill weight of grams. (Given that σ = 1.6 grams) Suppose that in a routine check of filling-process performance, intended to detect any change of the process mean from its target value, a sample of 25 jars produced an average of g. What does this value have to say about the plausibility of the current process mean actually being at the target of grams?

Generally Applicable Large-n Significance Tests for µ For observations that are describable as essentially equivalent to random selections with replacement from a single population with mean µ and variance σ 2, if n is large

Example A vendor claims that bottles produced by his manufacturing process have a mean internal strength of 150 psi. A potential customer thinks that the vendor is overstating the strength and selects a random sample of 36 bottles from the line. These bottles have a mean internal strength of 148 psi with a standard deviation of 5.5 psi. Is there enough evidence to refute the vendor’s claim?