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 #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 ?₁= mean TBV of patients that are considered normal. Let ?₂ = mean TBV of patients that had schizophrenia. Which of the following statements correctly defines the null hypothesis H_O?

A.  ?₁ + ?₂= 0

B.  ?₁ – ?₂< 0 (?₁ < ?₂)

C.  ?₁ ? ?₂ > 0 (?₁ > ?₂)

D.  ?₁ ? ?₂ = 0 (?₁ = ?₂)

Enter letter corresponding to correct answer

(ii)   Let ?₁= mean TBV of patients that are considered normal. Let ?₂ = mean TBV of patients that had schizophrenia. Which of the following statements correctly defines the alternate hypothesis H_A?

A.  ?₁ ? ?₂ = 0 (?₁ = ?₂)

B.  ?₁ – ?₂< 0 (?₁ < ?₂)

C.  ?₁ ? ?₂ > 0 (?₁ > ?₂)

D.  ?₁ + ?₂= 0

Enter letter corresponding to correct answer

(iii) Enter the level of significance ? used for this test:

Enter in decimal form. Examples of correctly entered answers:  0.01    0.02    0.05    0.10

(iv) For sample from population with mean = ?1 : Determine sample mean x??1 and sample standard deviation s1iv  For sample from population with mean = ?1 :          Determine sample mean x¯1 and sample standard deviation s1

Enter sample mean in integer form (no decimals), then comma, then sample standard deviation in integer form. Examples of correctly entered answers:  

13145,1211

-112845,13187

(v) For sample from population with mean = ?2 : Determine sample mean x??2 and sample standard deviation s2v  For sample from population with mean = ?2 :       Determine sample mean x¯2 and sample standard deviation s2

Enter sample mean in integer form (no decimals), then comma, then sample standard deviation in integer form. Examples of correctly entered answers:  

13145,1211

-112845,13187

(vi) Determine degrees of freedom df :

Enter value in decimal form rounded down to nearest whole number. Examples of correctly entered answers:

2 3 16 110

(vii) Determine test statistic:

Enter value in decimal form rounded to nearest ten-thousandth. Examples of correctly entered answers:

0.0104      3.0370        16.5000        110.0819

(viii) Using tables, calculator, or spreadsheet: Determine and enter p-value corresponding to test statistic.

Enter value in decimal form rounded to nearest ten-thousandth. Examples of correctly entered answers:

0.0001     0.0021     0.0305     0.6004      0.8143    1.0000

(ix) Comparing p-value and ? value, which is the correct decision to make for this hypothesis test?  

A. Reject H_o

B. Fail to reject H_o

C. Accept H_o

D. Accept H_A

Enter letter corresponding to correct answer.

  (x) Select the statement that most correctly interprets the result of this test:

A. The result is not statistically significant at .10 level of significance. Sufficient evidence exists to support the claim that patients with schizophrenia have less TBV on average than a patient that is considered normal.

B. The result is not statistically significant at .10 level of significance. There is not enough evidence to support the claim that patients with schizophrenia have less TBV on average than a patient that is considered normal.

C. The result is statistically significant at .10 level of significance. Sufficient evidence exists to support the claim that patients with schizophrenia have less TBV on average than a patient that is considered normal.

D. The result is statistically significant at .10 level of significance. There is not enough evidence to support the claim that patients with schizophrenia have less TBV on average than a patient that is considered normal.

Answer 1

Following is the output of descriptive statistics:

Descriptive statistics
	X1, normal patients	X2, schizophrenia patients
count	32	31
mean	14,63,339.22	14,51,293.19
sample standard deviation	1,25,458.28	1,71,932.23
sample variance	15,73,97,79,000.05	29,56,06,90,486.83
minimum	1203344	1066075
maximum	1688990	1861991
range	485646	795916

So we have

(i)

The null hypothesis is:

(ii)

The alternative hypothesis is

(iii)

The level of significance: 0.10

(iv-v)

(vi)

Here degree of feedom will be

The critical value for 61 degree of freedom and 0.10 level of signficance are: +/- 1.670

(vii)

The pooled standard deviation:

-------------

So standard error for difference in population mean is

The t-statistics will be

(viii)

The p-value is: 0.3756

(ix)

B. Fail to reject Ho

(x)

B. The result is not statistically significant at .10 level of significance. There is not enough evidence to support the claim that patients with schizophrenia have less TBV on average than a patient that is considered normal.