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Exam

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This article presents a comparative analysis of exam scores using boxplots, histograms, F-test, confidence interval and hypothesis test. The analysis includes construction of boxplots, histograms, F-test for variances, confidence interval and hypothesis test for the mean.

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STAT 3013

Institution Name

Student Name

Date of Submission

Institution Name

Student Name

Date of Submission

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1. Construction of the boxplots

When the two boxplots are compared its evident that the distributions of the scores do

differ. The median, minimum as well as the maximum point in each of the data set is

difference from the other.

The two data sets have no outliers as there is no indication from the boxplot

presented.

2. Histogram

Exam

1

bin

Frequenc

y

40 1

50 2

60 3

70 12

80 3

90 11

More 8

When the two boxplots are compared its evident that the distributions of the scores do

differ. The median, minimum as well as the maximum point in each of the data set is

difference from the other.

The two data sets have no outliers as there is no indication from the boxplot

presented.

2. Histogram

Exam

1

bin

Frequenc

y

40 1

50 2

60 3

70 12

80 3

90 11

More 8

40 50 60 70 80 90 More

0

2

4

6

8

10

12

14

Histogram

Frequency

bin

Frequency

Exam

2

bin

Frequenc

y

60 1

70 5

80 11

90 11

More 10

60 70 80 90 More

0

2

4

6

8

10

12

Histogram

Frequency

bin

Frequency

From the histograms developed for the scores in exam 1 and 2 the two scores do not

appear to follow a normal distribution. This is because if we plot a line curve through

the centre of the pillars it does not produce a bell-shaped graph.

3. F-test

0

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Histogram

Frequency

bin

Frequency

Exam

2

bin

Frequenc

y

60 1

70 5

80 11

90 11

More 10

60 70 80 90 More

0

2

4

6

8

10

12

Histogram

Frequency

bin

Frequency

From the histograms developed for the scores in exam 1 and 2 the two scores do not

appear to follow a normal distribution. This is because if we plot a line curve through

the centre of the pillars it does not produce a bell-shaped graph.

3. F-test

From the given data

Exam 1 Exam 2

32 55

45 62

50 63

56 66

58 67

60 67

61 71

61 73

63 74

64 74

64 74

65 75

66 76

67 77

67 77

68 78

69 80

69 82

72 83

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81 86

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85 87

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87 89

87 90

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89 91

90 92

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93 97

94 99

96

98

The F-test was used to test the hypothesis

Exam 1 Exam 2

32 55

45 62

50 63

56 66

58 67

60 67

61 71

61 73

63 74

64 74

64 74

65 75

66 76

67 77

67 77

68 78

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72 83

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81 86

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93 97

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96

98

The F-test was used to test the hypothesis

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H0 :Var 1=Var 2

Vs

H1 :Var 1 â‰ Var 2

From the excel output below the P value is 0.0135 which is less than 0.05. This falls

within the rejection region hence we reject the null hypothesis. We thereby conclude

that the variances of the two examinations are significantly difference.

F-Test Two-Sample for Variances

Exam 1 Exam 2

Mean 75.2250 81.4474

Variance 250.6404 120.3620

Observations 40 38

df 39 37

F 2.0824

P(F<=f) one-tail 0.0135

F Critical one-tail 1.7208

4. Confidence Interval

Vs

H1 :Var 1 â‰ Var 2

From the excel output below the P value is 0.0135 which is less than 0.05. This falls

within the rejection region hence we reject the null hypothesis. We thereby conclude

that the variances of the two examinations are significantly difference.

F-Test Two-Sample for Variances

Exam 1 Exam 2

Mean 75.2250 81.4474

Variance 250.6404 120.3620

Observations 40 38

df 39 37

F 2.0824

P(F<=f) one-tail 0.0135

F Critical one-tail 1.7208

4. Confidence Interval

Exam 1 Exam 2

Mean 75.225 Mean 81.44736842

Standard Error 2.503199875 Standard Error 1.779725141

Median 77 Median 83.5

Mode 61 Mode 74

Standard Deviation 15.83162609 Standard Deviation 10.97096258

Sample Variance 250.6403846 Sample Variance 120.3620199

Kurtosis -0.196045775 Kurtosis -0.545489604

Skewness -0.576070143 Skewness -0.47377318

Range 66 Range 44

Minimum 32 Minimum 55

Maximum 98 Maximum 99

Sum 3009 Sum 3095

Count 40 Count 38

Confidence Level(95.0%) 5.063199659 Confidence Level(95.0%) 3.606065666

Mean variation

Exam 1 Exam 2

70.16180034 80.28819966 77.84130275 85.05343409

From the constructed confidence interval, it can be noticed that the value of the mean

does overlap for the two examinations. It can therefore be deduced that the mean of

the two exam scores are equal and neither exam have a higher mean than the other

5. Hypothesis test for the mean

Using the data

Mean 75.225 Mean 81.44736842

Standard Error 2.503199875 Standard Error 1.779725141

Median 77 Median 83.5

Mode 61 Mode 74

Standard Deviation 15.83162609 Standard Deviation 10.97096258

Sample Variance 250.6403846 Sample Variance 120.3620199

Kurtosis -0.196045775 Kurtosis -0.545489604

Skewness -0.576070143 Skewness -0.47377318

Range 66 Range 44

Minimum 32 Minimum 55

Maximum 98 Maximum 99

Sum 3009 Sum 3095

Count 40 Count 38

Confidence Level(95.0%) 5.063199659 Confidence Level(95.0%) 3.606065666

Mean variation

Exam 1 Exam 2

70.16180034 80.28819966 77.84130275 85.05343409

From the constructed confidence interval, it can be noticed that the value of the mean

does overlap for the two examinations. It can therefore be deduced that the mean of

the two exam scores are equal and neither exam have a higher mean than the other

5. Hypothesis test for the mean

Using the data

Exam 1 Exam 2

32 55

45 62

50 63

56 66

58 67

60 67

61 71

61 73

63 74

64 74

64 74

65 75

66 76

67 77

67 77

68 78

69 80

69 82

72 83

76 84

78 85

81 86

83 86

83 87

85 87

86 88

87 89

87 90

88 91

89 91

90 92

90 93

91 93

92 94

92 94

93 95

93 97

94 99

96

98

The test conducted is used to test the hypothesis

H0 : Mean1=Mean2

Vs

32 55

45 62

50 63

56 66

58 67

60 67

61 71

61 73

63 74

64 74

64 74

65 75

66 76

67 77

67 77

68 78

69 80

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96

98

The test conducted is used to test the hypothesis

H0 : Mean1=Mean2

Vs

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H1 : mean1<mean 2

t-Test: Two-Sample Assuming Unequal Variances

Exam 1 Exam 2

Mean 75.225 81.44736842

Variance 250.6403846 120.3620199

Observations 40 38

Hypothesized Mean Difference 0

df 70

t Stat -2.025913408

P(T<=t) one-tail 0.023293331

t Critical one-tail 2.380807482

P(T<=t) two-tail 0.046586662

t Critical two-tail 2.647904624

From the excel output we deduced the P value as 0.02329 which is greater than 0.01.

This does not fall in the rejection region. We thereby fail to reject the null hypothesis

and conclude that the mean of the two examinations have no significance difference.

We are 97.67% confidence that the means of the two exams show no statistical

variations.

References

Bowerman, B., O'Connell, R. & Murphree, E., 2017. Business Statistics in Practice: Using Data,

Modeling, and Analytics. 8th ed. s.l.:McGraw-Hill Education.

t-Test: Two-Sample Assuming Unequal Variances

Exam 1 Exam 2

Mean 75.225 81.44736842

Variance 250.6403846 120.3620199

Observations 40 38

Hypothesized Mean Difference 0

df 70

t Stat -2.025913408

P(T<=t) one-tail 0.023293331

t Critical one-tail 2.380807482

P(T<=t) two-tail 0.046586662

t Critical two-tail 2.647904624

From the excel output we deduced the P value as 0.02329 which is greater than 0.01.

This does not fall in the rejection region. We thereby fail to reject the null hypothesis

and conclude that the mean of the two examinations have no significance difference.

We are 97.67% confidence that the means of the two exams show no statistical

variations.

References

Bowerman, B., O'Connell, R. & Murphree, E., 2017. Business Statistics in Practice: Using Data,

Modeling, and Analytics. 8th ed. s.l.:McGraw-Hill Education.

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