Each of 16 students measured the circumference of a tennis ball by four different methods, which were:

a: estimate the circumference by eye

b: measure the diameter with a ruler, then compute the circumference

c: measure the circumference with ruler and string

d: measure the circumference by rolling the ball along a ruler the results (in cm) are given below, in increasing order for each method:

a: 18.0, 18.0, 18.0, 19.0, 22.0, 22.0, 22.5, 22.9, 24.0, 24.0, 25.0, 25.0, 25.0, 25.0, 26.0, 26.4

b: 18.8, 18.9, 18.9, 19.8, 20.1, 20.4, 20.4, 20.4, 20.6, 20.5, 21.2, 21.9, 22.0, 22.0, 22.0, 23.6

c: 20.2, 20.5, 20.5, 20.7, 20.8, 20.9, 21.0, 21.0, 21.0, 21.0, 21.0, 21.5, 21.5, 21.5, 21.5, 21.6

d: 20.0, 20.0, 20.0, 20.2, 20.2, 20.5, 20.5, 20.7, 20.7, 20.7, 21.0, 21.5, 21.5, 21.6, 22.1, 22.3

Question Continuation

a. Compute the mean measurement for each method.

b. Compute the median measurement for. each method.

c. Compute the 20% trimmed mean measurement for each method.

a. Mean

Mean A = 22.675

Mean B = 20.71875

Mean C = 21.0125

Mean D = 20.86875

b. Median

Median A = 23.45

Median B = 20.45

Median C = 21.0

Median D = 20.7

c. 20% Trimmed Mean

Trimmed Mean A = 23.14

Trimmed Mean B = 20.73

Trimmed Mean C = 21.04

Trimmed Mean D = 20.75

Explanation:

a.

Mean = Σx/n

Where Σx= Summation of entries in each method

n = 16

Mean of Method A = (18.0 + 18.0 +18.0 + 19.0 + 22.0 + 22.0 + 22.5 + 22.9 + 24.0 + 24.0 + 25.0 +25.0 + 25.0 + 25.0 + 26.0 +26.4)/16

Mean A = 362.8/16

Mean A = 22.675

Mean of Method B = (18.8 + 18.9 + 18.9 + 19.8 + 20.1 + 20.4 + 20.4 + 20.4 + 20.6 + 20.5 + 21.2 + 21.9 + 22.0 + 22.0 + 22.0 + 23.6)16

Mean B = 331.5/16

Mean B = 20.71875

Mean of Method C = (20.2 + 20.5 + 20.5 + 20.7 + 20.8 + 20.9 + 21.0 + 21.0 + 21.0 + 21.0 + 21.0 + 21.5 + 21.5 + 21.5 + 21.5 + 21.6)/16

Mean C = 336.2/16

Mean C = 21.0125

Mean of Method D = (20.0 + 20.0 + 20.0 + 20.2 + 20.2 + 20.5 + 20.5 + 20.7 + 20.7 + 20.7 + 21.0 + 21.5 + 21.5 + 21.6 + 22.1 + 22.3)/16

Mean D = 333.9/16

Mean D = 20.86875

b.

Median is the number(s) at the middle

Since the total number of points for each method is 16 and this is an even number we need to calculate the median as the average between the 8th and the 9th position of the data ordered from the smallest to the largest. If we do this we have that:

Method A

Arranged Data: 18.0, 18.0, 18.0, 19.0, 22.0, 22.0, 22.5, 22.9, 24.0, 24.0, 25.0, 25.0, 25.0, 25.0, 26.0, 26.4

The 8th and 9th data are: 22.9 and 24.0

Median = (22.9 + 24.0/2

Median = 46.9/2

Median A = 23.45

Method B

Arranged Data: 18.8, 18.9, 18.9, 19.8, 20.1, 20.4, 20.4, 20.4, 20.5, 20.6, 21.2, 21.9, 22.0, 22.0, 22.0, 23.6

The 8th and 9th data are: 20.4 and 20.5

Median = (20.4+20.5)/2

Median = 40.9/2

Median B = 20.45

Method C

Arranged Data: 20.2, 20.5, 20.5, 20.7, 20.8, 20.9, 21.0, 21.0, 21.0, 21.0, 21.0, 21.5, 21.5, 21.5, 21.5, 21.6

The 8th and 9th data are: 21.0 and 21.0

Median = (21.0 + 21.0)/2

Median = 42.0/2

Median C = 21.0

Method D

Arranged Data: 20.0, 20.0, 20.0, 20.2, 20.2, 20.5, 20.5, 20.7, 20.7, 20.7, 21.0, 21.5, 21.5, 21.6, 22.1, 22.3

The 8th and 9th data are: 20.7 and 20.7

Median = (20.7+20.7)/2

Median = 41.4/2

Median D = 20.7

c.

The trimmed mean by 20% means that we'll calculate mean by removing the 20% from each of the tails.

20% of 16 = 3.2 (Approximated to 3)

So we need to remove 3 observations from both ends of the data.

Method A becomes

19.0, 22.0, 22.0, 22.5, 22.9, 24.0, 24.0, 25.0, 25.0, 25.0

Trimmed Mean A = (19.0+22.0+22.0+22.5+ 22.9+24.0+24.0+25.0+ 25.0+ 25.0)/10

Trimmed Mean A = 231.4/10

Trimmed Mean A = 23.14

Method B becomes

19.8, 20.1, 20.4, 20.4, 20.4, 20.6, 20.5, 21.2, 21.9, 22.0

Trimmed Mean B = (19.8+20.1+20.4+ 20.4+ 20.4+20.6+20.5+21.2+21.9+22.0)/10

Trimmed Mean B = 207.3/10

Trimmed Mean B = 20.73

Method C becomes

20.7, 20.8, 20.9, 21.0, 21.0, 21.0, 21.0, 21.0, 21.5, 21.5

Trimmed Mean C = (20.7+ 20.8+ 20.9+ 21.0+ 21.0+ 21.0+21.0+, 21.0+, 21.5+21.5)/10

Trimmed Mean C = 210.4/10

Trimmed Mean C = 21.04

Method D becomes

20.2, 20.2, 20.5, 20.5, 20.7, 20.7, 20.7, 21.0, 21.5, 21.5

Trimmed Mean D = (20.2+, 20.2+20.5+, 20.5+ 20.7+ 20.7+ 20.7 +, 21.0 +21.5 + 21.5)/10

Trimmed Mean D = 207.5/10

Trimmed Mean D = 20.75

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