Question

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 #3.1 contains the TBV of the normal patients and table #3.1 contains the TBV of schizophrenia patients ("SOCR data oct2009," 2013). 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.

Table #3.1: 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 #3.2: 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

Step 1: State the question

Step 2: Identify the information

1. State the random variables and parameters in words

2. State the null and alternative hypotheses and level of significance

3. Determine the type of test this is, then state and check the assumptions for this type of test

Step 3: Perform calculations

1. Find the sample statistics

2. Find the test statistic

3. Find the p-value

4. State the conclusion (compare p and alpha and decide to reject H_o or fail to reject H_o)

Step 4. Interpret (answer the original question in a complete sentence)

Please paste explanation from a word document

Answer 1

X = Total Brain Volume (in mm3) of Normal Patients.

Y = Total Brain Volume (in mm3) of Schizophrenia Patients.

X & Y be random variables are defined above.

Let   = Be the population mean of total Brain Volume (in mm3) of Normal Patients

and = be the population mean of total Brain Volume (in mm3) of Schizophrenia Patients

using this parameters we define the hypothesis.

A study was conducted that measured the total brain volume (TBV) (in mm3) of patients that had schizophrenia and patients that are considered normal.

Hypothesis:

Ho:The patients with schizophrenia have greater than or equal to TBV on average than a patient that is considered normal.

V/s

H1: The patients with schizophrenia have less TBV on average than a patient that is considered normal.

Symbolically:

The level of significance is 0.10 and it is used for the conclusion.

It is two-sample independent t-test

Under Ho,

test statistic is,

Assumption of this test is data should be normally distributed.

X	Y
1663407	1331777
1453510	1368378
1474790	1690182
1441636	1563593
1574808	1325525
1512571	1636119
1583940	1487886
1650348	1476891
1317156	1569413
1432033	1317885
1502702	1588573
1635918	1066075
1299470	1443775
1288971	1177002
1441045	1420249
1420416	1476254
1203344	1297327
1535137	1337827
1366346	1387893
1463498	1363859
1480171	1648209
1319737	1499983
1431890	1658258
1326402	1483763
1650207	1238979
1360810	1354054
1688990	1861991
1578698	1588132
1503005	1688950
1523045	1286638
1410213	1354649
1292641

Using minitab we check normality assumption using kolmogrov-smirnov test

and here p-value > alpha=0.10 for both RV then we conclude data is normally distributed.

Using Excel,

Excel => Data => Data Analysis => t-Test: Two-Sample Assuming Equal Variances => select variables => output range => ok

t-Test: Two-Sample Assuming Equal Variances
	X	Y
Mean	1463339.219	1451293.194
Variance	15739779000	29560690487
Observations	32	31
Pooled Variance	22536948584
Hypothesized Mean Difference	0
df	61
t Stat	0.31841
P(T<=t) one-tail	0.37563
t Critical one-tail	1.29558
P(T<=t) two-tail	0.75126
t Critical two-tail	1.67022

t stat=0.31841

p-value=0.3756 (# one tailed)

p-value >  alpha=0.10 we accept Ho and conclude that  the patients with schizophrenia have greater than or equal to TBV on average than a patient that is considered normal.

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