Module 3 Case Frequency Distribitions (Math Problem Sample)

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?