# Chi square test of independence, Confidence Interval

Problem 23: In one part of a test developed by a psychologist, the test subject is asked to form a word by unscrambling the letters "ciiatttsss". Given below are times (in seconds) required by 15 randomly selected persons to unscramble the letters. Construct a 95% confidence interval for standard deviation. Assume the population is approximately normal.

Times (in seconds)

68.7

27.4

26

60.5

34.6

61.1

68.6

48.4

43.6

39.5

85.3

26.3

43.4

83.7

68.9

Problem 24: Doing the "Statistics Modules" seems to help statistics students on their exams. Students in several statistics class are classified by grade in the course and by how many modules they completed. There wer 18 possible modules. The results are listed in the following table.

Grade in course 16 or more 9 to 15 8 or less

A 25 9 3

B 18 11 6

C 5 14 12

D or F 2 10 14

Does the data suggest that course grade and number of modules completed are dependent? Use an alpha=0.05 level of significance

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23 In one part of a test developed by a psychologist, the test subject is asked to form a word by unscrambling the letters "ciiatttsss". Given below are times (in seconds) required by 15 randomly selected persons to unscramble the letters. Construct a 95% confidence interval for standard deviation. Assume the population is approximately normal.

Times (in seconds)

68.7

27.4

26

60.5

34.6

61.1

68.6

48.4

43.6

39.5

85.3

26.3

43.4

83.7

68.9

First we calculate variance and standard deviation of the data

Score (Score-mean)^2

68.7 265.69000

27.4 625.00000

26 696.96000

60.5 65.61000

34.6 316.84000

61.1 75.69000

68.6 262.44000

48.4 16.00000

43.6 77.44000

39.5 166.41000

85.3 1082.41000

26.3 681.21000

43.4 81.00000

83.7 979.69000

68.9 272.25000

786.00 5664.64000

n= 15

mean= 52.4 =786/15

Variance=s 2 404.6171429 =5664.64/14

standard deviation= 20.1151 =square root of 404.617143

Find Chi square values for 95% confidence interval

Degrees of fredom=15-1=14

chi square vlaues

X upper ^2 = 26.11894805

X lower ^2 = 5.628726168

Variance Standard deviation=square root of Variance

s L 2 = (n-1) s ^2 / X upper ^2 = 216.87857 14.7268

s U 2 = (n-1) s ^2 / X lower ^2 = 1006.38045 31.7235

95% confidence interval

Lower limit of standard deviation 31.7235

Upper limit of standard deviation 14.7268

Problem 24

Doing the "Statistics Modules" seems to help statistics students on their exams. Students in several statistics class are classified by grade in the course and by how many modules they completed. There wer 18 possible modules. The results are listed in the following table.

Grade in course 16 or more 9 to 15 8 or less

A 25 9 3

B 18 11 6

C 5 14 12

D or F 2 10 14

Does the data suggest that course grade and number of modules completed are dependent? Use an alpha=0.05 level of significance

Null Hypothesis: Course grade and number of modules completed are independent

Alternative Hypothesis: Course grade and number of modules completed are dependent

Theoretical=Row total X column total/n

Thus for Grade A and >16 theoretical=37*50/129= 14.3411

and so on

No of modules completed

>16 9-15 <8

Grade in course Actual Theoretical Actual Theoretical Actual Theoretical Row total

A 25 14.3411 9 12.6202 3 10.0388 37

B 18 13.5659 11 11.938 6 9.4961 35

C 5 12.0155 14 10.5736 12 8.4109 31

D or F 2 10.0775 10 8.8682 14 7.0543 26

129

Column total 50 44 35

Chi square value= summation of (fo-fe)^2/fe= 36.9017

fo fe (fo-fe)^2/fe

25 14.3411 7.92214

18 13.5659 1.44931

5 12.0155 4.09615

2 10.0775 6.47442

9 12.6202 1.03848

11 11.938 0.0737

14 10.5736 1.11033

10 8.8682 0.14445

3 10.0388 4.93532

6 9.4961 1.28713

12 8.4109 1.53154

14 7.0543 6.83877

36.9017

No of degrees of freedom= (No of rows-1)*(No of columns-1)=(4-1)*(3-1)=6

alpha (a) = 0.05

Chi square value= 12.5916 from the tables

Accptance region for null hypothesis is upto 12.5916

Since Chi square value computed in this case= 36.9017

which lies to the right of acceptance region of 12.5916

Reject Null Hypothesis

Course grade and number of modules completed are dependent

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