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Math 10-Hypothesis Testing - comparing two means Name gw7 1. What is the difference between two samples that are dependent and two samples that are independent? Give an example of two dependent samples and two independent samples. 2. What conditions are necessary in order to use the dependent samples t-test for the mean of the difference of two populations? In Problems 3-10, classify the two given samples as independent or dependent. Explain your reasoning by identifying the 2 populations for independent sampling, or the one population and 2 measurements for dependent sampling. 3. Sample 1: The SAT scores for 35 high school students who did not take an SAT preparation course Sample 2: The SAT scores for 40 high school students who did take an SAT preparation course 4. Sample 1: The SAT scores for 44 high school students Sample 2: The SAT scores for the same 44 high school students after taking an SAT preparation course 5. Sample 1: The weights of 51 adults Sample 2: The weights of the same 51 adults after participating in a diet and exercise program for one month 6. Sample 1: The weights of 40 females Sample 2: The weights of 40 males 7. Sample 1: The average speed of 23 powerboats using an old hull design Sample 2: The average speed of 14 powerboats using a new hull design7. Sample 1: The average speed of 23 powerboats using an old hull design Sample 2: The average speed of 14 powerboats using a new hull design 8. Sample 1: The fuel mileage of 10 cars Sample 2: The fuel mileage of the same 10 cars using a fuel additive 9. The table shows the braking distances (in feet) for each of four different sets of tires with the car's anti-lock braking system (ABS) on and with ABS off. The tests were done on ice with cars traveling at 15 miles per hour Tire Set 1 2 3 Braking distance with ABS 42 55 43 61 Braking distance without ABS 58 67 59 75 10. The table shows the heart rates (in beats per minute) of five people before exercising and after. Person 1 2 3 5 Heart Rate before Exercising 42 55 13 61 65 Heart Rate after Exercising 58 67 59 75 9011. You work for the statistics department of the National Center for Health Patients 17 and Younger with Statistics. In an effort to determine if the mean length of hospital stays for Pneumonia (10 Years Ago) patients diagnosed with pneumonia is different today from what it was 10 years ago, your department decides to analyze data from a random selection of X = 4.3 hospital records. The results for several patients ages 17 and younger are shown $1 = 1.2 in the histograms for 10 years ago and this year. Your department believes that 1 =20 Frequency there is a difference in the mean length of hospital stays for patients diagnosed with pneumonia. As the department's research analyst, it is your job to work with the data and determine if this claim can be supported. 6 7 A. How Would You Do It? Length of stay (in days) (a) What sampling technique would you use to select a representative sample Patients 17 and Younger with for the study? Why? Pneumonia (This Year) 14 4 12 2 = 3.4 10 $7 = 0.7 n2 = 25 Frequency (b) What sampling technique would you use if you divided hospital records according to four geographic regions (northeast, south, midwest, and west), randomly obtained records from hospitals in that where the totals matched the proportion in each region? 2 3 4 5 6 Length of stay (in days) (c) Does this method produce a representative sample? Identify possible flaws or biases in your study.B. Choosinga Test To test the claim that there is a difference in the mean length of hospital stays for patients [ages 17 and younger} diagnosed with pneumonia, are the samples independent or dependent? Do you need to know what each population's distribution is? Do you need to know anything about the population variances? C Conducting the Test Can you support the claim that there is a difference in the mean length of hospital stays for patients {ages 17 and younger] diagnosed with pneumonia? Assume the populations are normal and the population variances are equal. Use ct = 0.05. State Ho and Ha in words and parameters What is the correct model? What assumptions are needed? What is the significance level of the test? What is the decision rule when you use the p-value method? Hypothesis Test Independent Groups (t-test, pooled variance} The results of the test are shown to the right. What is correct p- this . . . 1DYearsA o ear value? Do you reject or fall to FEJEEI: Ho? lLlL' 1 a ,. | .....,. c. What is the significance level of the test? d. What is the decision rule when you use the p-value method? Hypothesis Test: Independent Groups (t-test, pooled variance) e. The results of the test are shown to the right. What is correct p- this 10YearsAgo year value? Do you reject or fail to reject Ho? 4.3 3.4 mean std. 1.2 0.7 dev. 20 25 n 43 df. 0.9000 difference (10YearsAgo - this year) 0.9098 pooled variance 0.9538 pooled std. dev. 0.2861 standard error of difference 0 hypothesized difference f. State your conclusion. Has there been a significant change in 3.15 t hospital stay length? 0030 p-value (two-tailed) 9985 p-value (one-tailed, lower) 0015 p-value (one-tailed, upper)12. A random sample of 10 patients with mild atherosclerotic disease was selected. Two measurements were taken on each patient, the thickness of plaque (in tenths of micrometers) in their carotid artery before Vitamin E treatment and after two years of taking Vitamin E daily. The results are given in the table below. Patient 1 2 3 4 5 6 7 8 9 10 After 60 69 79 63 60 65 62 67 68 70 Treatment Before 66 72 85 62 59 63 64 70 73 68 Treatment We want to know if these data provide sufficient evidence at the 1% level of significance to conclude that the Vitamin E treatment decreases the thickness of plaque in the carotid artery. (a) (DESIGN) State your Hypothesis (e) (DATA) Identify the correct test and circle the p-value, and circle your decision. Matched Pairs t-tests Pooled Variance t-tests Test Test Null hypothesis Ho: H_difference = 0 Null hypothesis Ho: Hi - Hz = 0 Alternative hypothesis H1: p_difference 0 Alternative hypothesis Hi: Hi - H: > 0 T-Value P-Value T-Value DF P-Value -1.87 0.953 -0.64 18 0.735 (c) (DESIGN) Significance Level Reject Ho Fail to Reject Ho (f) (CONCLUSION) State your overall conclusion in (d) (DESIGN) Decision Rule (p-value language that is clear, relates to the original