Exercises - Run Test

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EXERCISES – GROUP 8

1. A supervisor records the number of employees absent over a 30- day period. Test the claim

that the number of employees absent occur at random, at α = 0.05.

27 6 19 24 18 12 15 17 18 20 0 9 4 12 3 2 7 7 0 5 32 16 38 31 27 15 5 9 4 10

2. Table below shows the actual daily occurrence of sunshine in Atlanta during November 1974,

as a percentage of the possible time the sun could have shone if it had not been for cloudy skies.

Dichotomize the observation according to whether the amount of sunshine was more than 50%

of possible or 50% less, and test the null hypotheses that the pattern of occurrences of the two

types of day is random.

Percentage of day during which sunshine occurred in Atlanta, November 1974.

Day Percentage

16 100

17 4618 7

19 12

20 54

21 87

22 100

23 100

24 88

Day Percentage

1 85

2 85

3 99

4 70

5 17

6 74

7 100

8 289 100

10 100

11 31

12 86

13 100

14 0

15 100



25 50

26 100

27 100

28 100

29 48

30 0

3. In an article on quality control, Puecell(E30) gives the set of typical data shown in table

below. Categorize each observation according to whether it falls above or below 1435, and test

the claim that the pattern of occurrences is at random.

Typical data for life of incandescent lamps in hours, before establishment of control.

Sample Median

17 1210

18 1620

19 1560

20 730

21 1260

Sample Median

1 1100

2 1280

3 1460

4 1350

5 1060

6 1250

7 1440

8 1230

9 1630

10 2100

11 1210

12 1760

13 2410

14 2080

15 1500

16 1550



22 1560

23 1770

24 1160

25 1300

26 1500

27 127028 1560

29 1150

30 1940

31 840

32 1140

4. Columns 1 and 2 of table below show, for 15 normal fetuses, the gestational age and mean Q-

A0 (a measurement of the cardiac cycle) values, as reported by Murata and Martin (E29). If we

perform a regression analysis on the data using gestational age as the residuals by subtracting X

and mean Q – A0 as the dependent variable Y, we obtain the residuals by subtracting the fitted

from the observed value Y (shown in column 3). Dichotomize the residuals according to whether

they are negative or positive, and test the claim that their pattern of occurrences is random.

Observed age, mean Q – A0 values, and residuals obtained by fitting a regression line to the

data.

Gestational age 40 39 40 38 40 40 39 37

Mean Q – A0 71.5 71.5 72.5 64.4 69.3 72.7 67.7 61.1Residual -1.4 +2.8 -0.4 -0.5 -3.6 -0.2 -1.2 +0.2

Gestational age 38 39 40 38 36 39 36

Mean Q – A0 69.5 69.5 71.8 68.3 57.5 70.7 51.6

Residual +4.6 +0.6 -1.1 +3.4 +0.6 +5.9 -6.5

Answer

EXERCISES – GROUP 8

1. A supervisor records the number of employees absent over a 30- day period. Test the claim

that the number of employees absent occur at random, at α = 0.05.

27 6 19 24 18 12 15 17 18 20 0 9 4 12 3 2 7 7 0 5 32 16 38 31 27 15 5 9 4 10



Solution.

Hypotheses:

H0 : The pattern of occurrences of number of employees absent occur at random.(claim)

H1: The pattern of occurrences of number of employees absent is not random.

Test statistics:

• Median = 15

• Run, r = 6 , n1=14, n2 = 14

Critical value:

• Lower critical value = 9

• Upper critical value = 21

Decision:

Since , we reject H0

Conclusion:

Enough evidence to reject the claim that the pattern of occurrences of number of employee

absent occur at random.

2. Table below shows the actual daily occurrence of sunshine in Atlanta during November 1974,

as a percentage of the possible time the sun could have shone if it had not been for cloudy skies.

Dichotomize the observation according to whether the amount of sunshine was more than 50%

of possible or 50% less, and test the null hypotheses that the pattern of occurrences of the two

types of day is random.

Percentage of day during which sunshine occurred in Atlanta, November 1974.



Day Percentage

16 100

17 46

18 7

19 12

20 54

21 87

22 100

23 100

24 88

25 50

26 100

27 100

28 100

29 48

30 0

Solution

Hypotheses:

H0: The pattern of occurrences of the two types of day is random.(claim)

H1: The pattern of occurrences of the two types of day is not random.

Test statistics:

Day Percentage

1 85

2 85

3 99

4 70

5 176 74

7 100

8 28

9 100

10 100

11 31

12 86

13 100

14 0

15 100



• Run, r = 14, n1 = 20, n2 = 10

Critical value:



Decision:

Since , do not reject H0 .

Conclusion:

Not enough evidence to reject the claim that the pattern of occurrences of the two types of day israndom.



3. In an article on quality control, Puecell (E30) gives the set of typical data shown in table

below. Categorize each observation according to whether it falls above or below 1435, and test

the claim that the pattern of occurrences is at random.

Typical data for life of incandescent lamps in hours, before establishment of control.

Sample Median

17 1210

18 1620

19 1560

20 730

21 1260

22 1560

23 1770

24 1160

25 1300

26 1500

27 1270

28 156029 1150

30 1940

31 840

32 1140

Solution

Sample Median

1 1100

2 1280

3 1460

4 1350

5 1060

6 1250

7 1440

8 1230

9 1630

10 2100

11 1210

12 1760

13 2410

14 2080

15 1500

16 1550



Hypotheses:

H0: The pattern of occurrences is at random.(claim)

H1: The pattern of occurrences is not random.

Test statistics:

• Run, r = 19, n1 = 16, n2 = 16

Critical value:



Decision:

Since , do not reject H0.

Conclusion:

Not enough evidence to reject the claim that the pattern of occurrences is at random.



4. Columns 1 and 2 of table below show, for 15 normal fetuses, the gestational age and mean Q-

A0 (a measurement of the cardiac cycle) values, as reported by Murata and Martin (E29). If we

perform a regression analysis on the data using gestational age as the residuals by subtracting X

and mean Q – A0 as the dependent variable Y, we obtain the residuals by subtracting the fitted

from the observed value Y (shown in column 3). Dichotomize the residuals according to whether

they are negative or positive, and test the claim that their pattern of occurrences is random.

Observed age, mean Q – A0 values, and residuals obtained by fitting a regression line to the

data.

Gestational age 40 39 40 38 40 40 39 37

Mean Q – A0 71.5 71.5 72.5 64.4 69.3 72.7 67.7 61.1

Residual -1.4 +2.8 -0.4 -0.5 -3.6 -0.2 -1.2 +0.2

Gestational age 38 39 40 38 36 39 36

Mean Q – A0 69.5 69.5 71.8 68.3 57.5 70.7 51.6

Residual +4.6 +0.6 -1.1 +3.4 +0.6 +5.9 -6.5

Solution

Hypotheses:

H0: The pattern of occurrences is random. (claim)

H1: The pattern of occurrences is not random.

Test statistics:

• Run, r = 7, n1 = 8, n2 = 7



Critical value:



Decision:

Since , do not reject H0.

Conclusion:

There is not enough evidence to reject the claim that the pattern of occurrences is at random.

Exercises - Run Test

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