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Lecture 23 – April 15, 2008

Overview Chapter 10 – Two Sample Tests
Comparing Two Means (ss unknown) Paired Comparisons

Chapter 10 – Comparing Two Independent Means (ss unknown)
Format of Hypotheses: Just as when the standard deviations are known, the hypotheses for comparing two independent population means m1 and m2 are:

Chapter 10 – Comparing Two Independent Means (ss unknown)
Test Statistic: If the population variances s12 and s22 are known, then we use a standard normal distribution (Tuesday’s notes). If the population variances s12 and s22 are unknown, then we use a student’s t distribution. There are three possible cases… Case 1: Known Variances (Last Lecture) Use the standard normal table to define the rejection region or calculate the p-value.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Case 2: Unknown Variances, Assumed Equal Since the variances are unknown, they must be estimated and the Student’s t distribution used to test the means. Assuming the population variances are equal, s12 and s22 can be used to estimate a common pooled variance sp2. The test statistic is With degrees of freedom n = n1 + n2 – 2.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Case 3: Unknown Variances, Assumed Unequal If the unknown variances are assumed to be unequal, they are not pooled together. In this case, the distribution of the random variable x1 – x2 is not certain. Use the Welch-Satterthwaite test which replaces s12 and s22 with s12 and s22 in the known variance Z* formula, then uses a Student’s t test with adjusted degrees of freedom.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Case 3: Unknown Variances, Assumed Unequal Welch-Satterthwaite test with degrees of freedom A Quick Rule for degrees of freedom is to use n ~ min(n1 – 1, n2 – 1).

Chapter 10 – Comparing Two Independent Means (ss unknown)
Steps in Testing Two Means: Choose the level of significance, a. Choose the appropriate hypotheses Calculate the Test Statistic State the decision rule – Based on a, determine the critical value(s). Make the decision – Reject H0 if the test statistic falls in the rejection region(s) as defined by the critical value(s). For example, for a two-tailed test for Student’s t and a = .05

Chapter 10 – Comparing Two Independent Means (ss unknown)
When to Use Each Case Statistic: If the sample sizes are equal, the Case 2 and Case 3 test statistics will be identical, although the degrees of freedom may differ. If the variances are similar, the two tests will usually agree. If no information about the population variances is available, then the best choice is Case 3. The fewer assumptions, the better – When in doubt, use Case 3! Must Sample Sizes Be Equal? Unequal sample sizes are common and the formulas still apply.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example: A restaurant chain is considering closing one of two stores. In a sample of 16 randomly selected days, restaurant A has average daily sales of \$3000 with a standard deviation of SA = \$450. In a sample of 12 randomly selected days, restaurant B has average daily sales of \$2700 with a standard deviation of SB = \$400. All other things being equal, the restaurant with lower sales will be closed. Assuming that sales for restaurants within the same chain will have equivalent standard deviations, test the appropriate hypothesis to determine whether sales at restaurant B are significantly lower than sales at restaurant A at the 5% level of significance. (Overhead)

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example (continued): Assumptions: Sales data are independent The standard deviations are unknown, but assumed equal. We will treat this as Case 2: Hypotheses: we want to determine if mA > mB… H0: mA < mB vs. H1: mA > mB (Right-tail test) Test Statistic: (using pooled variance sp2)…

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example (continued): Test Statistic: (using pooled variance sp2)… t distribution with n = – 2 = 26 d.f. under H0.

Clickers What is the rejection criteria for this problem?
(A) Reject H0 in favor of H1 if T* > (B) Reject H0 in favor of H1 if T* > (C) Reject H0 in favor of H1 if T* > (D) Reject H0 in favor of H1 if T* > (E) Reject H0 in favor of H1 if T* >

Clickers What is your decision? (A) Reject H0 in favor of H1.
(B) Fail to Reject H0 in favor of H1. (C) Not enough information.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example (conclusion): Since our test statistic, T*, falls in the rejection region, we will reject H0 in favor of H1. Based on the data collected, there is statistically significant evidence that mA > mB. So, all other things being equal, we will close restaurant B.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example (revisited): How does our test change if we don’t assume equal variances (Case 3)? We will use the same hypothesis test: H0: mA < mB vs. H1: mA > mB (Right-tail test) With a different test statistic… T distribution with n* ~ min(nA – 1,nB – 1) = min(16 – 1, 12 – 1) = 11 d.f. under H0.

Chapter 10 – Comparing Two Independent Means (ss unknown)
Example (revisited): Rejection Criteria: For the right-tail test, we will reject H0 in favor of H1 if T* > ta,n. Decision: Since T* = > ta,n = t.05,11 = 1.796, we will reject H0 in favor of H1. Just as before, based on the data collected, there is statistically significant evidence that mA > mB. Our exact p-value will be a little larger in this case since this test makes fewer assumption – and is therefore more conservative.

Chapter 10 – Paired Comparisons (Dependent Samples)
Paired Data: Data occurs in matched pairs when the same item is observed twice but under different circumstances. For example, blood pressure is taken before and after a treatment is given. Paired data are typically displayed in columns:

Chapter 10 – Paired Comparisons (Dependent Samples)
Paired t Test: Paired data typically come from a before/after experiment on n subjects – so the n observations of the two variable are dependent. In the paired t test, the difference between x1 and x2 is measured as d = x1 – x2. The mean d and standard deviation sd of the sample of n differences are calculated with the usual formulas for a mean and standard deviation.

Chapter 10 – Paired Comparisons (Dependent Samples)
Paired t Test: The calculations for the mean and standard deviation are: Since the population variance of d is unknown, use the Student’s t with n – 1 degrees of freedom.

Chapter 10 – Paired Comparisons (Dependent Samples)
Selection of H0 and H1: Remember, the conclusion we wish to test should be stated in the alternative hypothesis. Based on the problem statement, we choose from… H0: md > 0 H1: md < 0 (ii) H0: md < 0 H1: md > 0 (iii) H0: md = 0 H1: md 0 If the null hypothesis is true and md = 0, then T* has the student’s t distribution with n – 1 d.f. Decision Criteria: (the same as any other t-test) We can either compare T* to a critical value of the appropriate t distribution or calculate (bound) the p-value for the test.

Chapter 10 – Paired Comparisons (Dependent Samples)
Example: A new cell phone battery is being considered as a replacement for the current one. Six college students were selected to try each battery in their usual mix of “talk” and “standby” and to record the number of hours until recharge was needed. The data is below. Using a level of significance of a = 5%, do these results show that the newer battery has significantly longer life? Student 1 2 3 4 5 6 New Battery 41 53 40 43 49 Old Battery 34 38 44 33 di = New - Old 7 13 2 -1 5 10 (Overhead)

Chapter 10 – Paired Comparisons (Dependent Samples)
Example (continued): We can calculated the sample mean and standard deviation for the dis… and Based on the problem statement, we will test the hypothesis H0: md < 0 vs. H1: md > 0 (which corresponds to the battery life for the new battery being greater). The test statistic is… t distribution with n = n – 1 = 5 d.f. under H0. or

Chapter 10 – Paired Comparisons (Dependent Samples)
Example (continued): Rejection Criteria: For the right-tail test, we will reject H0 in favor of H1 if T* > ta,n. Decision: Since T* = 2.86 > ta,n = t.05,5 = 2.015, we will reject H0 in favor of H1. Based on the data collected, there is statistically significant evidence that the new battery lasts longer.

Clickers For this paired t test, our test statistic T* = 2.86 has
a student’s t distribution with n = 5 degrees of freedom. Use the t-table to find appropriate bounds on the p-value of this test. (A) 0.01 < p-value < 0.02 (B) 0.02 < p-value < 0.025 (C) < p-value < 0.05 (D) 0.05 < p-value < 0.10