Question 7 (1 point) Question 7 options: A study was conducted that measured the

Question

Question 7 (1 point)

Question 7 options:

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 s1

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 s2{

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:  tc

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

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

Explanation / Answer

Solution:

Part i

Here, we have to find the sample mean and sample standard deviation for the data from table 1.

Sample mean = X1bar = 1463339

Sample standard deviation = S1 = 125458

(Mean and standard deviation are calculated by using excel)

Part ii

Here, we have to find the sample mean and sample standard deviation for the data from table 2.

Sample mean = X2bar = 1451293

Sample standard deviation = S2 = 171932

(Mean and standard deviation are calculated by using excel)

Part iii

From given information, we have

Level of confidence = c = 90% = 0.90

Level of significance = ? = 1 – c = 1 – 0.90 = 0.10

Level of significance = 0.10

Part iv

Sample size of first sample = N1 = 32

Sample size of second sample = N2 = 31

Degrees of freedom = N1 + N2 – 2 = 32 + 31 – 2 = 61

Degrees of freedom = 61

Part v

We have

Degrees of freedom = 61

Level of significance = 0.10

So, by using t-table or excel, we have

Critical value = 1.670

Part vi

Error bound of the mean = E = margin of error = Interval half width

E = Critical value * sqrt[Sp2*((1/n1)+(1/n2))]

Where Sp2 is pooled variance

Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

Sp2 = [(32 – 1)* 125458.2759^2 + (31 – 1)* 171932.2264^2]/(32 + 31 – 2)

Sp2 = 22536948583.7099

E = Critical value * sqrt[Sp2*((1/n1)+(1/n2))]

E = 1.670*sqrt[22536948583.7099*((1/32)+(1/31))]

E = 63188.1356

Error bound of the mean = E = 63188

Part vii

Confidence interval = (X1bar – X2bar) -/+ E

Confidence interval = (1463339.219 - 1451293.194) -/+ 63188

Confidence interval = 12046.0252 -/+ 63188

Lower limit = 12046.0252 - 63188 = -51142.1104

Upper limit = 12046.0252 + 63188 = 75234.1608

-51142.1104 < (µ1 - µ2) < 75234.1608

-51142 < (µ1 - µ2) < 75234

Part viii

From given alternatives, most correct alternative is given as below:

We estimate with 90% confidence that the true mean TBV of people considered normal is anywhere from 51565 mm^3 less to 75677 mm^3 more than the mean TBV for people with schizophrenia.

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