Module 3 Case Frequency Distribitions

TITLE OF PROJECT
By:
[student’s name]
[course]
[university]
[date]
1 The math grades on the final exam varied greatly. Using the scores below, how many scores were within one standard deviation of the mean? How many scores were within two standard deviations of the mean?
1 34   86   57   73   85    91   93   46    96   88   79   68   85   89
To answer this item, first we need to compute for standard deviation. We will use this formula.
We will compute this formula in 4 steps.
Step 1: Find the mean.
Mean = (99+34+86+57+73+85+91+93+46+96+88+79+68+85+89) / 15
Mean = 77.9333.
Step 2: Create the following table.
data

data-mean

(data - mean)2

99

21.0667

443.80584889

34

-43.9333

1930.13484889

86

8.0667

65.07164889

57

-20.9333

438.20304889

73

-4.9333

24.33744889

85

7.0667

49.93824889

91

13.0667

170.73864889

93

15.0667

227.00544889

46

-31.9333

1019.73564889

96

18.0667

326.40564889

88

10.0667

101.33844889

79

1.0667

1.13784889

68

-9.9333

98.67044889

85

7.0667

49.93824889

89

11.0667

122.47184889

Step 3: Find the sum of numbers in the last column to get.
Step 4: Calculate σ using the above formula.
Using the mean and standard deviation of these scores we can now get the z-score for each grade using this formula:
Z value = (X - µ) / σ
Where, X = Standardized Random Variable,
µ = Sample Mean,
σ = Sample Standard Deviation.
The table below summarizes the z-score for each math grade
Math grade

99

34

86

57

73

85

91

93

46

96

88

79

68

85

89

z-score

1.11

-2.31

0.42

-1.1

-0.26

0.37

0.69

0.79

-1.68

0.95

0.53

0.06

-0.52

0.37

0.58

There are 11 math grades that are within one (1) standard deviation away from the mean while 14 math grades are within two (2) standard deviations away from the mean. All math grades are within three (3) standard deviations away from the mean.
1 The scores for math test #3 were normally distributed. If 15 students had a mean score of 74.8% and a standard deviation of 7.57, how many students scored above an 85%?
Result
P (X>85)=0.0885
Explanation
Step 1: Sketch the curve.
The probability that X>85 is equal to the blue area under the curve.
Step 2:
Since μ=74.8 and σ=7.57 we have:
P ( X>85 )=P ( X‒μ>85‒74.8 )=P ( X‒μσ>85‒74.87.57)
Since Z=x‒μσ and 85‒74.87.57=1.35 we have:
P ( X>85 )=P ( Z>1.35 )
Step 3: Use the standard normal table to conclude that:
P (Z>1.35)=0.0885
0.0885 x 100 = 8.85%
Only 8.85% of the class (or one person) scored above 85% for Math test #3.
2 If you know the standard deviation, how do you find the variance?
Standard deviation is actually just the square root of the variance. Hence, if you know the standard deviation, just square it and you will be able to get the variance.
3 To get the best deal on a stereo system, Louis called eight appliance stores and asked for the cost of a specific model. The prices he was quoted are listed below:
$216   $135   $281   $189   $218   $193   $299   $235
Find the standard deviation.
The standard deviation of the data set is: σ=52.2597
Explanation
To find standard deviation we use the following formula
We will compute this formula in 4 steps.
Step 1: Find the mean. (X¯¯¯)
In this example: X¯¯¯=220.75.
Step 2: Create the following table.
data

data-mean

(data - mean)2

216

-4.75

22.5625

135

-85.75

7353.0625

281

60.25

3630.0625

189

-31.75

1008.0625

218

-2.75

7.5625

193

-27.75

770.0625

299

78.25

6123.0625

235

14.25

203.0625

Step 3: Find the sum of numbers in the last column to get.
Step 4: Calculate σ using the above formula.
1 A company has 70 employees whose salaries are summarized in the frequency distribution below.
Salary

Number of Employees

5,001–10,000

8

10,001–15,000

12

15,001–20,000

20

20,001–25,000

17

25,001–30,000

13



To answer this item, first we need to find the midpoint of the salary ranges. The table below summarizes the information need to compute for next steps.
Salary

Midpoint Salary

Number of Employees

5,001–10,000

7,500.5

8

10,001–15,000

12,500.5

12

15,001–20,000

17,500.5

20

20,001–25,000

22,500.5

17

25,001–30,000

27,500.5

13

* Find the standard deviation
To compute for the standard deviation of the grouped data, first we will use this formula:
‚[ (Σ(mi-μ...