Question

Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected.

• Researcher 1: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34

• Researcher 2: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29

Compute the mean, median, mode, range, and standard deviation for the data sets for each researcher.

Answer 1

For researcher 1 the data set is 3, 4, 11, 15, 16, 17, 22, 44, 37, 16, 14, 24, 25, 15, 26, 27, 33, 29, 35, 44, 13, 21, 22, 10, 12, 8, 40, 32, 26, 27, 31, 34, 29, 17, 8, 24, 18, 47, 33, 34.

Mean:

The mean is calculated as:

Mean = (3 + 4 + 11 + 15 + 16 + 17 + 22 + 44 + 37 + 16 + 14 + 24 + 25 + 15 + 26 + 27 + 33 + 29 + 35 + 44 + 13 + 21 + 22 + 10 + 12 + 8 + 40 + 32 + 26 + 27 + 31 + 34 + 29 + 17 + 8 + 24 + 18 + 47 + 33 + 34)/40
= 943/40
Mean = 23.575

Median:

Median is the middle value of the dataset when arranged in increasing order since the sample size is n =40 hence the median value is between 20th and 21st position.

The data set is arranged in ascending order as:

3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47

Thus the median value is (24 + 24)/2 = 24

Mode:

Mode is the most occuring value hence we can see that 33, 27, 8, 15, 16, 17, 22, 44, 34, 29, 24, 26, each appeared 2 times hence there will be multiple modes mode.

Range:

Range of the data set is calculated as Largest value - Smallest value

=> Range = 47-3 = 44

Standard deviation:

The standard deviation is calculated as:

= √(1/40 - 1) x ((3 - 23.575)² + ( 4 - 23.575)² + ...................... + ( 33 - 23.575)² + ( 34 - 23.575)²)

= √(1/39) x ((-20.575)² + (-19.575)² + ......................................+ (9.425)² + (10.425)²)

= √(0.0256) x ((423.330625) + (383.180625) + ..........................+ (88.830625) + (108.680625))

= √(0.0256) x (4943.775)

= √(126.56064)

= 11.2589

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For researcher 2 the dataset is 3, 14, 11, 5, 16, 17, 28, 41, 31, 18, 14, 14, 26, 25, 21, 22, 31, 2, 35, 44, 23, 21, 21, 16, 12, 18, 41, 22, 16, 25, 33, 34, 29, 13, 18, 24, 23, 42, 33, 29

Mean:

The mean is calculated as:

Mean = (3 + 14 + 11 + 5 + 16 + 17 + 28 + 41 + 31 + 18 + 14 + 14 + 26 + 25 + 21 + 22 + 31 + 2 + 35 + 44 + 23 + 21 + 21 + 16 + 12 + 18 + 41 + 22 + 16 + 25 + 33 + 34 + 29 + 13 + 18 + 24 + 23 + 42 + 33 + 29)/40
= 911/40
Mean = 22.775

Median:

Median is the middle value of the dataset when arranged in increasing order since the sample size is n =40 hence the median value is between 20th and 21st position.

The data set is arranged in ascending order as:

2, 3, 5, 11, 12, 13, 14, 14, 14, 16, 16, 16, 17, 18, 18, 18, 21, 21, 21, 22, 22, 23, 23, 24, 25, 25, 26, 28, 29, 29, 31, 31, 33, 33, 34, 35, 41, 41, 42, 44

Thus the median value is (22 + 22)/2 = 24

Mode:

Mode is the most occuring value hence we can see that 21, 16, 18, 14, each appeared 3 times hence 21, 16, 18, 14 are the modes.

Range:

Range of the data set is calculated as Largest value - Smallest value

=> Range = 44-2 = 42

Standard deviation:

The standard deviation is calculated as:

= √(1/40 - 1) x ((3 - 22.775)² + ( 14 - 22.775)² + ............................................+ ( 33 - 22.775)² + ( 29 - 22.775)²)

= √(1/39) x ((-19.775)² + (-8.775)² + .................................................................................+ (10.225)² + (6.225)²)

= √(0.0256) x ((391.050625) + (77.000625) + ......................................................+ (104.550625) + (38.750625))

= √(0.0256) x (4200.975)

= √(107.54496)

= 10.3787

Note : Feel free to ask if query remains