Compare Two Different Groups of Individuals

We now address the second of the two two-sample situations. In this situation, we compare two di erent groups of individuals, with each group coming from a di erent population. Let 1 = mean of the rst population. Let 2 = mean of the second population. Let X 1 = mean of sample from rst population. Let X 2 = mean of sample from second population. Let n1 = size of sample from rst population. Let n2 = size of sample from second population. Let s1 = standard deviation of sample from rst population. Let s2 = standard deviation of sample from second population. It turns out that 1 X  (X s 2 ) (1 s21 s22 n1 + n2 2 ) e tdf where df = the smaller of (n1 - 1) and (n2 - 1). Exercise 1. Does cocaine use by pregnant women cause their babies to have low birth weight? To study this question, birth weights of babies of women who tested positive for cocaine/crack during a drug-screening test were compared with the birth weights for women who either tested negative or were not tested, a group we call \other". Here are the summary statistics. The birth weights are measured in grams. n Positive Test Other x 2733 3118 134 5974 s 599 672 Formulate appropriate hypotheses and carry out the test of signi cance for these data. Let 1 = average weight of all crack babies. Let 2 = average weight of all non-crack babies. H0 : 1 2 = - Let X 1 = 0. H1 : 1 2 < - 0. average weight of crack babies in a sample of size 134 . Let X 2 = The p average weight of non-crack babies in a sample of size 5974 . 1 value = P(X X 2  2733 3118 j Ho is true) 1 = P(X X 2  2733 3118 j 1 = P( 1 (X = P(t133 X ) (1 r2 s21 n1 + 3118) q  (2733 599 672 2 134 = P(t133 2 ) s22 n2  + 2 5974 7:337379536) 1 2 = 0)  (2733 0 ) 3118) r s21 n1 + (1 s22 n2 2 ) ) = Since 9:65
10 tcdf( 10; 7:337379536; 133) = 9:65
10 12 12 : < 0.05, we can reject Ho . There is enough evidence that the babies whose mothers took cocaine weigh less than the babies whose mothers didn't take cocaine. Exercise 2. The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this disease has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two elds are under study. The rst step in the research project is to compare the mean soil water content for the two elds (Source: Journal of Agricultural, Biological, and Environmental Statistics, Vol. 2, No. 2). Units are percent water by volume of soil. (#24) Field A samples, x1 : 10.2, 10.7, 15.5, 10.4, 9.9, 10.0, 16.6, 15.1, 15.2, 13.8, 14.1, 11.4, 11.5, 11.0 Field B samples, x2 : 8.1, 8.5, 8.4, 7.3, 8.0, 7.1, 13.9, 12.2, 13.4, 11.3, 12.6, 12.6, 12.7, 12.4, 11.3, 12.5 (i) Use a calculator with mean and standard deviation keys to verify that x1 2.39, x2  10.77, and s2 = 2.40.  12.53, s1  (ii) Assuming the distribution of soil water content in each eld is mound-shaped and symmetric, use a 5% level of signi cance to test the claim that eld A has, on average, a higher soil water content than eld B. 2 Class Exercise 1. Form 2 of the Gates-MacGintie Reading Test was administered to both an experimental group and a control group after 6 weeks of instruction, during which the experimental group received peer tutoring and the control group did not. For the experimental group n1 = 30 children, the mean score on the vocabulary portion of the test was x1 = 368.4, with sample standard deviation s1 = 39.5. The average score on the vocabulary portion of the test for the n2 = 30 subjects in the control group was x2 = 349.2, with sample standard deviation s2 = 56.6. Use a 1% level of signi cance to test the claim that the experimental group performed better than the control group. Answer: p-value = 0.0692 (#22) Class Exercise 2. This problem is based on information regarding productivity in leading Sil- icon Valley companies. In large corporations, an \intimidator" is an employee who tries to stop communication, sometimes sabotages others, and, above all, likes to listen to him- or herself talk. Let x1 be a random variable representing productive hours per week lost by peer employees of an intimidator. x1 : 8 3 6 2 2 5 2 A "stressor" is an employee with a hot temper that leads to unproductive tantrums in corporate society. Let x2 be a random variable representing productive hours per week lost by peer employees of a stressor. x2 : 3 3 10 7 6 2 5 8 i. Use a calculator with mean and standard deviation keys to verify that x1 = 4.00, s1  2.38, x2 = 5.5, s2  2.78. ii. Assuming the variables x1 and x2 are independent, do the data indicate that the population mean time lost due to stressors is greater than the population mean time lost due to intimidators? Use a 5% level of signi cance. (Assume the population distributions of time lost due to intimidators and time lost due to stressors are each mound-shaped and symmetric. Answer: p-value  0.1516 (#26) Class Exercise 3. The mean ACT score for 43 male high school students is 21.1 and the standard deviation is 5.0. The mean ACT score for 56 female high school students is 20.9 and the standard deviation is 4.7. At = 0.01, can you reject the claim that male and female high school students have equal ACT scores? Answer: p-value = 0.840 Class Exercise 4. A restaurant association says that households in the United States headed by people under the age of 25 spend less on food away from home than households headed by people ages 55-64. The mean amount spent by 30 households headed by people under the age of 25 is $2,015 and the standard deviation is $113. The mean amount spent by 30 households headed by people ages 55-64 is $2,045 and the standard deviation is $97. At = 0.05, can you support the restaurant association'sclaim? Answer:p-value= 0.139 3