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APA
Subject:
Mathematics & Economics
Type:
Math Problem
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English (U.S.)
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Topic:
Module 3 Case Frequency Distribitions (Math Problem Sample)
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All problems need to include all required steps and answers and all answers need to be reduced to the lowest terms. Please see instructions.
source..Content:
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 34Â Â Â 86Â Â Â 57Â Â Â 73Â Â Â 85Â Â Â 91Â Â Â 93Â Â Â 46Â Â Â 96Â Â Â 88Â Â Â 79Â Â Â 68Â Â Â 85Â Â Â 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:Â 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:Â 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
PÂ (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:
PÂ (Â X>85Â )=PÂ (Â Z>1.35Â )
Step 3:Â Use the standard normal table to conclude that:
PÂ (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?
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