Hypothesis Testing about a Population Mean

Hypothesis Testing Fundamentals Title: Hypothesis Testing about a Population Mean When not stated, α = .05 Key Concepts: For a test with fixed sample size, n, and given significance level, α, we draw a rejection region as a guideline for when the null hypothesis may be rejected. Examples of Rejection Regions Example 1: H0: μ = μ0 (claimed pop mean) Ha: μ < μ0 Diagram of left-tailed test, with shaded region labeled area = α Example 2: H0: μ = μ0 Ha: μ > μ0 Diagram of a right-tailed test, with shaded region labeled α * Ex: H0: μ= 10, Ha: μ < 10 Hypothesis Test Setup: Suppose we have a sample of n = 49 whose population standard deviation is σ = 20 and we wish to test the following at α = 0.07: H0: μ = 55 Ha: μ < 55 Test and Questions: This is a z-test. For what values of z will we reject the null hypothesis? For what values of x̅ will we reject the null hypothesis? What is the value of the probability of the type I error? Page 2: More Hypothesis Testing Additional Questions: Now suppose that the population mean has been determined to be μ = 47. What is P(Type II error)? P(H0 is NOT rejected | μ = 47) What is the probability of the Power? Popper 19: "At the bakery where you work, loaves of bread are supposed to weigh 1 pound, with standard deviation σ=0.13 pounds. You believe that new personnel are producing loaves that are lighter than 1 pound. As supervisor of Quality Control, you want to test your hypothesis at the 95% confidence level. You weigh 20 loaves and obtain a mean weight of 0.95 pounds." Questions and Answers: This is a ____ test. a. z The probability of a type 1 error is b. .05 The null and alternate hypothesis would be b. H0: μ = 1 Ha: μ < 1 Rejecting a true null hypothesis is classified as a a. type I error Failing to reject a false null hypothesis is classified as a b. type II error The level of significance for a hypothesis test is equal to the probability of a a. Type I error Additional Rejection Region Example: H0: μ = μ0 Ha: μ ≠ μ0 (different) Diagram of a two-tailed test, with both ends shaded, labeled α/2 each. Page 3: Test Statistics and P-value Test Statistics for μ: Normal Population, σ known: z = (x̅ - μ0) / (σ / √n) * pop. st. dev. Normal Population, σ unknown: t = (x̅ - μ0) / (s / √n) P-value definition Def: The P-value is the probability, calculated assuming that the null hypothesis is true, of obtaining a value of the test statistic at least as contradictory to H0 as the value calculated from the available sample. P(our sample yielded given results | H0 is true) Proposition: The P-value is the smallest significance level α at which the null hypothesis can be rejected. Thus, the P-value is sometimes referred to as the observed significance level (OSL) for the data. Example: "The tar content of a certain type of cigarette has been averaging 11.5 milligrams per cigarette with a standard deviation of 0.6 milligram. A researcher has discovered a new filter that he claims will reduce the mean tar content from 11.5. That is, he claims that if μ is the mean tar content of cigarettes with the new filter, μ < 11.5. "We consider n = 50 randomly selected cigarettes having the new filter and find that the average tar content is x̅ = 11.4. Is this enough evidence to reject H0 : μ = 11.5 at the α = 0.10 significance level?" x̅ = 11.4 (less?) diagram showing if the test is left tailed or not.