CH9: Testing the Difference Between Two Means, Two ...

1 CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 343 CH9: Testing the Difference Between Two Means, Two proportions , and Two Variances CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 344 Section 9-1 Testing the Difference Between Two Means: Using the Z Test Suppose we are interested in determining if a certain medication relieves patients headaches. We give the drug/treatment to one group and give a placebo to a control group and compare the mean incidences of patient relief from the headache Between the two groups. If the treatment group had a statistically significant improvement in headache symptoms over the control group, then we can conclude the drug works. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 345 So our question might be, Is the mean incidence of headache relief different for the two groups? Let 1 mean headache relief from treatment group and 2 mean headache relief from control group.

2 Then our hypotheses would be: H0: H1: Alternatively, we could state the hypotheses as: H0: H1: CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 346 Assumptions for the Test to Determine the Difference Between Two Means The samples must be independent of each other. That is, there can be no relationship Between the subjects in each sample. The populations from which the samples come must be (approximately) normally distributed or the sample sizes of both groups should be at least 30. The standard deviations of both populations must be known. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 347 We can compare the groups by the Difference in their population means, 1 2, where 1 is the population mean for group 1 and 2 is the population mean for group 2. We estimate 1 2 with x 1 x 2 The standard deviation of x 1 x 2 is 12n1 22n2 When both populations are normally distributed or the samples size for each group is at least 30, then x 1 x 2 has a normal distribution.

3 CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 348 Formula for the z test for Comparing Two Means from independent Populations H0: 1 2 k (or k or k) Note: We often k 0, but it doesn t have to be. Test value: z* (x 1 x 2) ( 1 2) 12n1 22n2 (x 1 x 2) k 12n1 22n2 CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 349 The observed Difference Between the sample means may be due to chance, in which case the null hypothesis will not be rejected. If the Difference is statistically significant, the null hypothesis is rejected and the researcher can conclude the population means are different. The same approach to finding critical values and P-values that was used in Section 8-2 will be used here (Table E or Table F with = ). CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 350 Example: Dr. Cribari would like to determine if there is a statistically significant Difference Between her two Math 2830 classes.

4 To make this comparison she will compare the results from exam 1. Class one had 35 students take the exam with a mean of and a population standard deviation of Class two had 32 students take the exam with a mean of 84 and a population standard deviation of Can Dr. Cribari conclude that there is Difference in the mean test grades Between the two classes at = Ho: 1 = 2 Ho: 1 2 Step 1 State the hypotheses and identify the claim. 012112 CLAIM:: HH CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 351 Step 2 Find the critical value(s) from the appropriate table. As stated, the problem is giving the population standard deviations. This means that we will be doing a z-test. Two-sided test critical value = = Step 3 Compute the test value and determine the P-value. *121222221212() ()( 84) 0 p-value = 2*.0202 = CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 352 Step 4 Make the decision to reject or not reject the null hypothesis.

5 Since the p-value is smaller than our , the null hypothesis is rejected. [OR, Since, our test value, , falls within the rejection region, the null hypothesis is rejected] Step 5 Summarize the results. That is, there is evidence to support the claim that the exam 1 grades differ Between the two sections of MATH2830. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 353 Example: A survey found that the average hotel room in New Orleans is $ and the average room rate in Phoenix is $ Assume that the data were obtained from two samples of 50 hotels each and that the (population) standard deviations were $ and $ , respectively. At = , can it be concluded that the average hotel room in New Orleans costs more than in Phoenix? Step 1 State the hypotheses and identify the claim. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 354 Step 2 Find the critical value(s) from the appropriate table.

6 Step 3 Compute the test value and determine the P-value. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 355 Step 4 Make the decision to reject or not reject the null hypothesis. Step 5 Summarize the results. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 356 Formula for the z Confidence Interval for Difference Between Two Means Assumptions: 1. The data for each group are independent random samples. 2. The data are from normally distributed populations and/or the sample sizes of the groups are greater than 30. 3. The population standard deviation is (assumed) known. Formula: Note: When n1 and n2 are at least 30, then s1 and s2 can be used in place of 1 and 2. 221212/ 212()xxznn CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 357 Example: Two brands of cigarettes are selected, and their nicotine content is compared.

7 The data are shown below. Find the 95% confidence interval of the true Difference in the means. Brand A Brand B X1 mg X2 mg 1 mg 2 mg n1 30 n2 40 22221212/ ()( ) ( ) ( , )xxznn CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 358 At , is there convincing evidence that the mean amount of nicotine differs Between the brands? CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 359 Example: For the hotel example, construct a 98% confidence interval of the true Difference in the : Testing the Difference Between Two Means or Two proportions Santorico - Page 360 Section 9-2: Testing the Difference Between Two Means of independent Samples: Using the t Test Many times the conditions set forth by the z test in Section 9-1 cannot be met ( , the population standard deviations are not known).

8 In these cases, a t test is used to test the Difference Between means when the two samples are independent and when the samples are taken from two normally or approximately normally distributed populations. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 361 Formula for the t Test for Testing the Difference Between Two Means: independent Samples. Variances are assumed to be unequal: t (X1 X2) ( 1 2)s12n1 s22n2 where degrees of freedom is equal to the smaller of n1 1 or n2 1. We will use Table F to find our critical values and our p-values. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 362 WARNING: Your calculator will perform a 2 sample t-test (its #4 under STATS then TESTS). However, it uses a complicated formula to determine the degrees of freedom that will ultimately affect how the calculator deals with confidence intervals and p-values.

9 We will come back to this point at the end of the section. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 363 Example: A real estate agent wishes to determine whether tax assessors and real estate appraisers agree on the values of homes. A random sample of the two groups appraised 10 homes. Is there a significant Difference in the values of the homes for each group? Let = Assume the data are from normally distributed populations. Real Estate Appraisers Tax Assessors X1 $83,256 X2 $88,354 s1 $3256 s2 $2341 n1 10 n2 10 Step 1 State the hypotheses and identify the claim. H0: 1= 2 H1: 1 2 Sample standard deviations given! Use a t-test CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 364 Step 2 Find the critical value(s) from the appropriate table. T-test means use the t-table (Table F).

10 We have 9 degrees of freedom since n1=10 and n2=10. The smallest of n1-1 and n2-1 is 9. Information we need: two-tailed test, = , df=9 T critical value is Step 3 Compute the test value and determine the P-value. 121222221212() ()(83, 256 88, 354) (0) CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 365 tailedt* = (t) 2 ( )2( ) CRITICAL REGION CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 366 Step 4 Make the decision to reject or not reject the null hypothesis. The null hypothesis is rejected. This decision can be based on: the fact that the test value ( ) is within the critical region since it is less than or the fact that the p-value ( ) is smaller than = Step 5 Summarize the results. There is significant evidence that tax assessors and real estate appraisers disagree on the values of homes. CH9: Testing the Difference Between Two Means or Two proportions Santorico - Page 367 Example: A researcher suggests that male nurses earn more than female nurses.