MA131 Lecture For a fixed sample size, α and β cannot be lowered simultaneously.

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1 Type I Error: α = P (H 0 is rejected H 0 is true). The value of α represents the significance level of the test. Type II Error: β = P (H 0 is not rejected H 0 is false). The value of 1 β is called the power of the test. About α and β... The two types of error depend on each other. For a fixed sample size, α and β cannot be lowered simultaneously. α and β are inversely proportional. We can decrease both α and β simultaneously by increasing the sample size. In most hypothesis testing, it is not easy to compute β. A table that summarizes the type of errors. Actual Situation H 0 is true H 0 is false Decision Do not reject Correct Type II or H 0 decision β error Reject Type I or Correct H 0 α error decision Tails of a Test The partition of the total region into rejection and nonrejection regions depends on α, the significance level of the test. A test with two rejection regions is called two tailed test, and a test with one rejection region is called a one tailed test. The one tailed test is called: a left tailed test if the rejection region is in the left tail of the distribution curve; a right tailed test if the rejection region is in the right tail of the distribution curve. c 2004, RSHavea, MaCS, USP 1 File updated: September 24, 2004

2 Example 1 (A two tailed test) It is reported that the mean family size in the United States was 3.18 in A researcher wants to check whether or not this mean has changed since Let µ be the current mean family size for all families. The two possible decisions are: 1. The mean family size has not changed, i.e. µ = The mean family size has changed, i.e. µ H 0 : µ = 3.18 H 1 : µ 3.18 The sign in H 1 determines what test we take. Example 2 (A left tailed test) A soft drink company claims that on average the amount of soda in any can is 12 oz. Suppose a consumer agency wants to test whether the mean amount of soda per can is less than 12 oz. Let µ be the mean amount of soda in all cans. The two possible decisions are: 1. The mean amount of soda in all cans is not less than 12 oz, i.e. µ = 12 ounces. 2. The mean amount of soda in all cans is less than 12 oz, i.e. µ < 12 oz. H 0 : µ = 12 H 1 : µ < 12 Note that we could have written H 0 : µ 12 and it will not affect the result of the test as the sign in H 1 is less than (<). c 2004, RSHavea, MaCS, USP 2 File updated: September 24, 2004

3 When the sign in H 1 is less than (<), as in this example, the test is always left tailed. Example 3 (A right tailed test) It is reported that the mean starting salary of school teachers in the US was $25,735 during It is required to test whether the current mean starting salary of all school teachers in the US is higher than $25,735. Let µ be the current mean starting salary of school teachers in the US. The two possible decisions are: 1. The mean starting salary of all school teachers in the US is not higher than $25,735, i.e. µ = $ The current mean starting salary of all school teachers in the US is higher than $25,735, i.e. µ > $25, 735. H 0 : µ = $25, 735 H 1 : µ > $25, 735 Note, we could have written H 0 : µ $25, 735 and it will not affect the conclusion. When the sign in H 1 is greater than (>), as in this example, the test is always right tailed. c 2004, RSHavea, MaCS, USP 3 File updated: September 24, 2004

4 Two tailed Left tailed Right tailed test test test Sign in H 0 = = or = or Sign in H 1 < > Rejection In both In the left In the right region tails tail tail Traditional Method: Step 1. State the hypotheses and identify the claim. Step 2. Find the critical value(s). Step 3. Compute the test value. Step 4. Make the decision. Step 5. Summarize the results. Hypothesis Tests About µ: large sample The z test for a mean In tests of hypotheses about µ for large samples, the test statistic is the random variable where z = X µ σ/ n, X = sample mean µ = hypothesized population mean σ = population deviation n = sample size Example 4 A researcher reports that the average salary of assistant professors (AP) is more than $42,000. A sample of 30 AP has a mean salary of $43,260. At α = 0.05, test the claim that AP earn more than $42,000/yr. It is known that σ = $5, 230. Step 1. State the hypotheses and identify the claim. H 0 : µ $42, 000 H 1 : µ > $42, 000 (claim) c 2004, RSHavea, MaCS, USP 4 File updated: September 24, 2004

5 Step 2. Find the critical value. Since α =.05 and the test is right tailed, C.V. = Step 3 Compute the test value. Graphically: z = X µ σ/ n = 43, , 000 5, 230/ 30 = Step 4. Make the decision. Since the test value, z = 1.32 is less than the critical value z = 1.65, and is to the left of the rejection region, the decision is: Do not reject H 0 Step 5. Summary. There is not enough evidence to support the claim that assistant professors earn more on average than $42,000. c 2004, RSHavea, MaCS, USP 5 File updated: September 24, 2004

9-7: THE POWER OF A TEST

CD9-1 9-7: THE POWER OF A TEST In the initial discussion of statistical hypothesis testing the two types of risks that are taken when decisions are made about population parameters based only on sample