# Case Frequency Distributions (Math Problem Sample)

Case Assignment

By submitting this assignment, you affirm that it contains all original work, and that you are familiar with Trident University’s Academic Integrity policy in the Trident Policy Handbook. You affirm that you have not engaged in direct duplication, copy/pasting, sharing assignments, collaboration with others, contract cheating and/or obtaining answers online, paraphrasing, or submitting/facilitating the submission of prior work. Work found to be unoriginal and in violation of this policy is subject to consequences such as a failing grade on the assignment, a failing grade in the course, and/or elevated academic sanctions. You affirm that the assignment was completed individually, and all work presented is your own.

Problems need to include all required steps and answer(s) for full credit. All answers need to be reduced to lowest terms where possible.

Answer the following problems showing your work and explaining (or analyzing) your results.

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?

99 34 86 57 73 85 91 93 46 96 88 79 68 85 89

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%?

If you know the standard deviation, how do you find the variance?

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.

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

Find the standard deviation.

Find the variance.

6. Calculate the mean and variance of the data. Show and explain your steps. Round to the nearest tenth.

14, 16, 7, 9, 11, 13, 8, 10

Create a frequency distribution table for the number of times a number was rolled on a die. (It may be helpful to print or write out all of the numbers so none are excluded.)

3, 5, 1, 6, 1, 2, 2, 6, 3, 4, 5, 1, 1, 3, 4, 2, 1, 6, 5, 3, 4, 2, 1, 3, 2, 4, 6, 5, 3, 1

Answer the following questions using the frequency distribution table you created in No. 7.

Which number(s) had the highest frequency?

How many times did a number of 4 or greater get thrown?

How many times was an odd number thrown?

How many times did a number greater than or equal to 2 and less than or equal to 5 get thrown?

The wait times (in seconds) for fast food service at two burger companies were recorded for quality assurance. Using the data below, find the following for each sample.

Range

Standard deviation

Variance

Lastly, compare the two sets of results.

Company Wait times in seconds

Big Burger Company 105 67 78 120 175 115 120 59

The Cheesy Burger 133 124 200 79 101 147 118 125

What does it mean if a graph is normally distributed? What percent of values fall within 1, 2, and 3, standard deviations from the mean?

Module 3 â€“ Case Frequency Distributions

Name:

Institution:

Response to question 1

Math grades: 99 34 86 57 73 85 91 93 46 96 88 79 68 85 89

Mean

Mean = Sum of math grades (X) / N (Number of students)

Mean= (99+34+86+57+73+85+91+93+46+96+88+79+68+85+89)/15

Mean=77.93333

Standard Deviation

To get standard deviation we find the square root of variance

S=â€‘X-M2n-1

Step 1: calculation of variance

Subtract the mean grade from each of the values

99-77.93333=21.06667

34-77.93333=-43.93333

86-77.93333=8.06667

57-77.93333=-20.93333

73-77.93333=-4.93333

85-77.93333=7.06667

91-77.93333=13.06667

93-77.93333=15.06667

46-77.93333=-31.93333

96-77.93333=18.06667

88-77.93333=10.06667

79-77.93333=1.06667

68-77.93333=-9.93333

85-77.93333=7.06667

89-77.93333=11.06667

Next you square all answers from subtraction above

(21.06667)2=443.80458

(-43.93333)2=1930.13748

(8.06667)2=65.07116

(-20.93333)2=438.20430

(-4.93333)2=24.33774

(7.06667)2=49.93782

(13.06667)2=170.73786

(15.06667)2=227.00454

(-31.93333)2=1019.73756

(18.06667)2=326.40456

(10.06667)2=101.33784

(1.06667)2=1.13778

(-9.93333)2=98.67104

(7.06667)2=49.93782

(11.06667)2=122.47118

Then add all the squared numbers

(443.80458+1930.13748+65.07116+438.20430+24.33774+49.93782+170.73786+227.00454+

1019.73756+326.40456+101.33784+1.13778+98.67104+49.93782+122.47118=5068.93326

Then divide the sum of squares by (n-1)

Where n=15, n-1 is 15-1=14

5068.93326/14=362.06666

Variance=362.06666

Step 2 calculate standard deviation

Standard deviation is the square root of variance

Standard deviation (S)= 362.06666

S=19.02805

* Scores within one standard deviation of the mean

Mean=77.93333

To get scores within one standard deviation 77.93333 +1 /-1

77.93333-1=76.93333

77.93333+1=78.93333

There is no score within one standard deviation of the mean. The nearest score is 79 but falls outside the range of 76.93333 and 78.93333

* Scores within two standard deviations of the mean

To get scores within two standard deviations 77.93333 +2 /-2

77.93333-2=75.9333

77.93333+2=79.93333

There is only one score within two standard deviations of the mean that is 79

Response to question 2

Mean=74.8%

Standard deviation=7.57

* Approximation of students who scored above 85%

Mean score=74.8%

Standard deviation of the marks =7.57

First step: Z-score for 85%

z = (X - Î¼) / Ïƒ

(85%-74.8%)/7.57=1.35

Second step: P (Z<1.35) =0.9115

Third step: subtract this from 1

1-0.9115=0.0885

Probability that some students scored above 85% is 0.0885

Students who scored above 85%

0.0885*15=1.3275

Round off since there is no fraction of a student

Two students scored above 85%.

* finding variance when given the standard deviation

If you are given the standard deviation, you simply square it to get the variance

Standard deviation2=variance

Variance=7.572

Variance=57.3049

Response to question 3

Standard Deviation

To get standard deviation we find the square root of variance

S=â€‘X-M2n-1

Step 1: calculation of variance

Subtract the mean grade from each of the values

Mean = ($216+ $135+$281+ $189+ $218+ $193+ $299+ $235)/8

Mean= $220.75

$216-$220.75=-$4.75

$135-$220.75=-$85.75

$281-$220.75=$60.25

$189-$220.75=-$31.75

$218-$220.75=-$2.75

$193-$220.75=-$27.75

$299-$220.75=$78.25

$235-$220.75=$14.25

Next you square all answers from subtraction above

(-$4.75)2=$22.5625

(-$85.75)2=$7353.0625

($60.25)2=$3630.0625

(-$31.75)2=$1008.0625

(-$2.75)2=$7.5625

(-$27.75)2=$770.0625

($78.25)2=$6123.0625

($14.25)2=$203.0625

Then add all the squared numbers

($22.5625+$7353.0625+$3630.0625+$10...

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