Question

For your doctoral dissertation, you decide to study the social well-being of 100,000 people. You ask them to complete a questionnaire that measures social well-being. Higher numbers indicate more positive well-being. Your results indicate that the average well-being score is 100 points with a standard deviation of 20 points. Use this information to answer problems 15-29.

15. What percent of the people had well-being scores higher than 125 points?

16. What percent of the people had well-being scores higher than 80 points?

17. What percent of the people had well-being scores less than 85 points?

18. What percent of the people had well-being scores less than 120 points?

19. What percent of the people had well-being scores between 90 and 110 points?

20. What percent of the people had well-being scores between 112 and 126 points?

21. What percent of the people had well-being scores between 75 and 90 points?

22. How many participants had a well-being score less than 115 points?

23. How many participants had a well-being score above 110 points?

24. How many participants had a well-being score between 100 and 120 points?

25. What well-being score represents the 75 rd percentile?

26. What well-being score represents the 33rd percentile?

27. What well-being score represents the 45th percentile?

28. What well-being score represents the 90th percentile?

29. What well-being score represents the 50th percentile?

30. What well-being score represents the 66th percentile?

Answer 1

(15)

= 100

= 20

To find P(X>125):

Z = (125 - 100)/20 = 1.25

Table of Area Under Standard Normal Curve gives area = 0.3944

So, P(X>125) = 0.5 - 0.3944 = 0.1056

So,

Answer is:

0.1056

(16)

To find P(X>80):

Z = (80 - 100)/20 = - 1.00

Table of Area Under Standard Normal Curve gives area = 0.3413

So, P(X>80) = 0.5 + 0.3413 = 0.8413

So,

Answer is:

0.8413

(17)

To find P(X<85):

Z = (85 - 100)/20 = - 0.75

Table of Area Under Standard Normal Curve gives area = 0.2734

So, P(X<85) = 0.5 - 0.2734 = 0.2266

So,

Answer is:

0.2266

(18)

To find P(X<120):

Z = (120 - 100)/20 = 1.00

Table of Area Under Standard Normal Curve gives area = 0.3413

So, P(X<120) = 0.5 + 0.3413 = 0.8413

So,

Answer is:

0.8413

(19)

To find P(90 < X < 110):

Case 1: For X from 90 to mid value:

Z = (90 - 100)/20 = - 0.50

Table of Area Under Standard Normal Curve gives area = 0.1915

Case 2: For X from mid value to 110:

Z = (110 - 100)/20 = 0.50

Table of Area Under Standard Normal Curve gives area = 0.1915

So, P(90 <X< 110) = 2 X 0.1915 = 0.3830

So,

Answer is:

0.3830

(20)

To find P(112 < X < 126):

Case 1: For X from mid value to 112:

Z = (112 - 100)/20 = 0.60

Table of Area Under Standard Normal Curve gives area = 0.2257

Case 2: For X from mid value to 126:

Z = (126 - 100)/20 = 1.30

Table of Area Under Standard Normal Curve gives area = 0.4032

So, P(112 <X< 126) = 0.2257 + 0.4032 = 0.6289

So,

Answer is:

0.6289

(22)

To find P(X<115):

Z = (115 - 100)/20 = 0.75

Table of Area Under Standard Normal Curve gives area = 0.2734

So, P(X<120) = 0.5 + 0.2734 = 0.7734

So,

Answer is:

0.7734

(23)

To find P(X>110):

Z = (110 - 100)/20 = 0.50

Table of Area Under Standard Normal Curve gives area = 0.1915

So, P(X>125) = 0.5 - 0.1915 = 0.3085

So,

Answer is:

0.3085

(24)

To find P(100<X<120):

Z = (120 - 100)/20 = 1.00

Table of Area Under Standard Normal Curve gives area = 0.3413

So, P(100<X<120) = 0.3413

So,

Answer is:

0.3413

(25)

75th percentile corresponds to area = 0.75 - 0.50 = 0.25 from mid value to Z on RHS.

Table gives Z = 0.675

So,

Z = 0.675 = (X - 100)/20

So,

X = 100 + (0.675 X 20) = 113.5

So,

Answer is:

113.5

(26)

33rd percentile corresponds to area = 0.50 -0.33 = 0.17 from mid value to Z on LHS.

Table gives Z = - 0.44

So,

Z = - 0.44 = (X - 100)/20

So,

X = 100 - (0.44 X 20) = 91.2

So,

Answer is:

91.2

(27)

45th percentile corresponds to area = 0.50 - 0.45 = 0.05 from mid value to Z on LHS.

Table gives Z = - 0.125

So,

Z = - 0.125 = (X - 100)/20

So,

X = 100 - (0.125 X 20) = 97.5

So,

Answer is:

97.5

(28)

90th percentile corresponds to area = 0.90 - 0.50 = 0.40 from mid value to Z on RHS.

Table gives Z = 1.28

So,

Z = 1.28 = (X - 100)/20

So,

X = 100 + (1.28 X 20) = 125.6

So,

Answer is:

125.6

(29)

50th Percentile = Mean = 100

So,

Answer is:

100

(30)

66th percentile corresponds to area = 0.66 - 0.50 = 0.16 from mid value to Z on RHS.

Table gives Z = 0.42

So,

Z = 0.42 = (X - 100)/20

So,

X = 100 + (0.42 X 20) = 108.4

So,

Answer is:

108.4