Q6A A study was conducted that measured the total brain volume (TBV) (in mm3) of patients that had schizophrenia and patients that are considered normal.  Table #1 contains the TBV of the normal patients and Table #2 contains the TBV of schizophrenia patients ("SOCR data Oct2009," 2013).  Table #1: Total Brain Volume (in mm3) of Normal Patients 1663407 1583940 1299470 1535137 1431890 1578698 1453510 1650348 1288971 1366346 1326402 1503005 1474790 1317156 1441045 1463498 1650207 1523045 1441636 1432033 1420416 1480171 1360810 1410213 1574808 1502702 1203344 1319737 1688990 1292641 1512571 1635918         Table #2: Total Brain Volume (in mm3) of Schizophrenia Patients 1331777 1487886 1066075 1297327 1499983 1861991 1368378 1476891 1443775 1337827 1658258 1588132 1690182 1569413 1177002 1387893 1483763 1688950 1563593 1317885 1420249 1363859 1238979 1286638 1325525 1588573 1476254 1648209 1354054 1354649 1636119             Is there enough evidence to show that the patients with schizophrenia have less TBV on average than a patient that is considered normal?  Test at the 10% level. (i)  Let μ1 = mean TBV of patients that are considered normal.  Let μ2 = mean TBV of patients that had schizophrenia.  Which of the following statements correctly defines the null hypothesis HO? A.  μ1 + μ2 = 0 B.  μ1 – μ2 < 0  (μ1 < μ2) C.  μ1 − μ2 > 0  (μ1 > μ2) D.  μ1 − μ2 = 0  (μ1 = μ2) Enter letter corresponding to correct answer    (ii)   Let μ1 = mean TBV of patients that are considered normal.  Let μ2 = mean TBV of patients that had schizophrenia.  Which of the following statements correctly defines the alternative hypothesis HA? A.  μ1 − μ2 = 0  (μ1 = μ2) B.  μ1 – μ2 < 0  (μ1 < μ2) C.  μ1 − μ2 > 0  (μ1 > μ2) D.  μ1 + μ2 = 0   Enter letter corresponding to correct answer     (iii)  Enter the level of significance α used for this test: Enter in decimal form.  Examples of correctly entered answers:  0.01    0.02    0.05    0.10

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...