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

Let ?₁= mean TBV of patients that are considered normal. Let ?₂ = mean TBV of patients that had schizophrenia. Compute a 90% confidence interval for the difference in TBV of normal patients and patients with Schizophrenia.

(i) For the sample from population with mean = ?1 : Determine sample mean x¯1 and sample standard deviation s1i  For the sample from population with mean = ?1 :      Determine sample mean x¯1 and sample standard deviation s1 {"version":"1.1","math":"<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle mathsize="14px"><mrow><mfenced><mi mathvariant="bold">i</mi></mfenced><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">For</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">the</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">from</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">population</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">with</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">mean</mi><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">=</mo><mo mathvariant="bold">&#xA0;</mo><msub><mi mathvariant="bold-italic">&#x3BC;</mi><mn mathvariant="bold">1</mn></msub><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">:</mo><mspace linebreak="newline"></mspace><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">Determine</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">mean</mi><mo mathvariant="bold">&#xA0;</mo><msub><mover><mi mathvariant="bold">x</mi><mo mathvariant="bold">&#xAF;</mo></mover><mn mathvariant="bold">1</mn></msub><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">and</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">standard</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">deviation</mi><mo mathvariant="bold">&#xA0;</mo><msub><mi mathvariant="bold-italic">s</mi><mn mathvariant="bold">1</mn></msub></mrow></mstyle></math>"}

Enter sample mean to nearest integer, then comma, then sample standard deviation to nearest integer. Examples of correctly entered answers:

1332,112

-15478,3651

   (ii) For the sample from the population with mean = ?2 : Determine sample mean x¯2 and sample standard deviation s2ii  For the sample from the population with mean = ?2 :      Determine sample mean x¯2 and sample standard deviation s2 {"version":"1.1","math":"<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle mathsize="14px"><mrow><mfenced><mi mathvariant="bold">ii</mi></mfenced><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">For</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">the</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">from</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">the</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">population</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">with</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">mean</mi><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">=</mo><mo mathvariant="bold">&#xA0;</mo><msub><mi mathvariant="bold-italic">&#x3BC;</mi><mn mathvariant="bold">2</mn></msub><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">:</mo><mspace linebreak="newline"></mspace><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">Determine</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">mean</mi><mo mathvariant="bold">&#xA0;</mo><msub><mover><mi mathvariant="bold">x</mi><mo mathvariant="bold">&#xAF;</mo></mover><mn mathvariant="bold">2</mn></msub><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">and</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">sample</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">standard</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="bold">deviation</mi><mo mathvariant="bold">&#xA0;</mo><msub><mi mathvariant="bold-italic">s</mi><mn mathvariant="bold">2</mn></msub></mrow></mstyle></math>"}

Enter sample mean to nearest integer, then comma, then sample standard deviation to nearest integer. Examples of correctly entered answers:

1332,112

-15478,3651

(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) Determine degrees of freedom df :

Enter value rounded DOWN to nearest integer.

(v) Determine t - score associated with critical value: t_c

Enter in decimal form to nearest thousandth. Examples of correctly entered answers:

0.0011    0.020     0.500     0.371 2.000

(vi) Determine "error bound of the mean" E

Enter value rounded to nearest integer.

(vii) Determine confidence interval estimate of the difference ?₁ – ?₂:

Enter lower bound value to nearest integer, followed by < , followed by "?1-?2" for difference, followed by <, followed by upper bound value to nearest integer. No spaces between any characters. Do not use italics, but if mathematically necessary use negative signs. Examples of correctly entered answers:

0.77<?1-?2<0.78

13.12<?1-?2<13.94

-9.73<?1-?2<-8.08

(viii) Using the confidence interval, select the most correct description of the result of the survey:

A. We estimate with 90% confidence that the mean TBV of people considered normal is anywhere from 51565 mm³ more to 75677 mm³ less than the mean TBV for people with schizophrenia.

B. We estimate with 90% confidence that the true proportional TBV of people considered normal is anywhere from 51565 mm³ less to 75677 mm³ more than the mean TBV for people with schizophrenia.

C. We estimate with 90% confidence that the sample mean TBV of people considered normal is anywhere from 51565 mm³ less to 75677 mm³ more than the mean TBV for people with schizophrenia.

D. We estimate with 90% confidence that the true mean TBV of people considered normal is anywhere from 51565 mm³ less to 75677 mm³ more than the mean TBV for people with schizophrenia.

Enter letter corresponding to most correct answer

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

(i-ii)

So we have

(iii)

(iv)

Here degree of feedom will be

(v)

The critical value for 61 degree of freedom and 90% confidnece interval is: 1.670

(vi)

The pooled standard deviation:

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

So standard error for difference in population mean is

The error bound is:

(vii)

The point estimate is:

The confidence interval is:

D. We estimate with 90% confidence that the true mean TBV of people considered normal is anywhere from 51565 mm3 less to 75657 mm3 more than the mean TBV for people with schizophrenia.