Question

2. Consider the following sample of exam scores, arranged in increasing order. DO IT BY HAND AND SHOW WORK. NO WORK NO CREDIT. MAKE SURE YOU USE THE FORMULAS IN CLASS NOTES TO FIND LOCATION FOR PERCENTILES/QUARTILES. THE TEXTBOOK USES A DIFFERENT APPROACH AND RESULTS WILL BE DIFFERENT!

   28 57 58 64 69 74 79 80 83 85 85 87 87 89 89 90 92 93 94 94 95 96 96 97 97 97 97 98 100 100

   (Hint: the sample mean of these exam scores is 85)

1. Determine the interquartile range (IQR).

2. Obtain the five number summary.

3. Identify potential outliers if any.

4. Construct and interpret a boxplot or if appropriate, a modified boxplot.

4. Using the data in question 3 above, use Excel to

1. Determine the frequency distribution of exam scores. Use cutpoint grouping with

   25 as the first cutpoint and classes of equal width 5.

2. Determine the relative frequency distribution of exam scores

3. Find the following descriptive statistics of the exam scores using excel functions:

   Sample size (count), Mean, Variance, Standard Deviation, Minimum, Maximum, Range, Summation of observations in sample (sum), Mode, Median, Lower quartile-Q1, Middle quartile-Q2 (same as median), upper quartile-Q3 and Inter- quartile Range (IQR).

Answer 1

Data in increasing order:

28 57 58 64 69 74 79 80 83 85 85 87 87 89 89 90 92 93 94 94 95 96 96 97 97 97 97 98 100 100

interquartile range (IQR) = Q₃ - Q₁

Where

Q₃ = Value of observation.

= value of (23.25)^th  observation.

= 23 ^th observation + 0.25 * (24 ^th obs - 23^th obs)

= 96 + 0.25 *(97-96)

= 96.25

Q₁ = Value of observation

= value of (7.75) th observation

= 7 ^th observation + 0.75 * (8 ^th obs - 9^th obs)

= 79 + 0.75 * (80-79)

= 79.75

IQR = 96.25 - 79.75 = 16.5

Five number summary:

Minimum = 28

Q₁ = 79.75

Q 2 = 92.5

Q3 = 96.25

Maximum = 100

Q₂ = Value of observation

=  value of (17.5) th observation

= 17 ^th observation + 0.5 * (18 ^th obs - 17^th obs)

= 92 + 0.5 * (93 - 92)

= 92.5

Identifying outliers:

The observation fall below Q1 - 1.5* IQR = 79.75 - 1.5 * 16.5 = 55 and above Q3 + 1.5 * IQR = 96.25 + 1.5 * 16.5 = 121

The value 28 is fall below 55 hence it is outlier.