Question: Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8408 1374 1872 8879 2459 11413 608 14138 6452 1850 2818 1356 10498 7478 4019 4341 739 2127 3653 5794 8305 a. Provide a five-number summary. b. Compute the lower and upper limits. c. Do the data contain any outliers? d. Johnson & Johnson’s sales are the largest on the list at $14,138 million. Suppose a data entry error (a transposition) had been made and the sales had been entered as $41,138 million. Would the method of detecting outliers in part (c) identify this problem and allow for correction of the data entry error? e. Show a box plot

 Step 1

a)

Concept Used:

The five-number summary is the lowest value, Quartile-1, median, Quartile-3 and highest value in any data set.

The median is the middle value of the data distribution when arranged in ascending order. The Quartile-1 is the middle value of lowest value and median of data distribution. The Quartile-3 is the middle value of highest value and median of data distribution.

Calculation:

The lowest value of the data distribution is 608 and highest value is 14138.

The median is the middle value of the data when arranged in ascending order which is 11^th value

  Median=4019

First quartile is average of 5^th and 6^th values

   Quartile−1=1850+18722=1861

Third Quartile is average of 16^th and 17^th values

   Quartile−3=8305+84042=8354.5

Conclusion:

The five number summary for annual sales of 21 pharmaceutical companies is 608, 1861, 4019, 8354.5, 14138.

Step 2

 (b)

Concept Used:

The formula for boundaries of outliers are,

  Lower Boundary=Q1−1.5*IQR

        Upper Boundary=Q3+1.5*IQR

The difference between third quartile and first quartile is Inter Quartile range.

Calculation:

Substituting the values in Inter quartile range formula,

  Inter Quartile​​​ Range=Q3−Q1=8354.5−1861=6493.5

Substituting in the boundaries of outliers formula,

  Lower Boundary=Q1−1.5*IQR=1861−1.5*6493.5=−7880.25

Upper Boundary=Q3+1.5*IQR

                           =8354.5+1.5*6493.5=18094.75

Conclusion:

The lower and upper limits for outliers of annual sales for 21 pharmaceutical companies are

-7880.25 and 18094.75, respectively.

 (c)

The lower and upper limits for outliers of annual sales for 21 pharmaceutical companies is -7880.25 and 18094.75. All data points in the given data are within the boundaries of outliers. Hence, we can say there are no outliers in the data.