The table below show Business statistics I examination marks for a class of one hundred students .

Marks 10-19 (4 student )

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20-29 ( 8 student )

30-39 ( a – student )

40-49 ( 22 student )

50-59 ( 48 student )

60-69 ( b – student )

The mean score of the students was 46.5

(i) Show that the values of a and b are 12 and 6 respectively. (5marks)

(ii) Estimate the median for the above sample. (3marks)

(iii) Determine the variance and the standard deviation (4 marks)

(iv) Given that 32% of the students failed the examination, what was the cut off points.

(3 marks)

(v)Estimate the mode for this data. (3marks)

i)

mean:

“\mu=\frac{\sum x_i}{n}=\frac{14.5\cdot4+24.5\cdot8+34.5a+44.5\cdot22+54.5\cdot48+64.5b}{4+8+a+22+48+b}=46.5”

“n=4+8+a+22+48+b=100”

Then:

“a+b=18”

“3849+34.5a+64.5b=4650”

“34.5(18-b)+64.5b=801”

“30b=180”

“b=180/30=6,\ a=18-6=12”

ii)

“median=l + [(n/2u2212c)/f] u00d7 h”

where,

- l = lower limit of median class
- n = total number of observations
- c = cumulative frequency of the preceding class
- f = frequency of median class
- h = class size

cumulative frequency:

10-19: 4

20-29: 12

30-39: 24

40-49: 46

50-59: 94

60-69: 100

median class: 50-59

“median=50+[(100/2-46)/94]\cdot9=50.38”

iii)

variance:

“\sigma^2=\frac{\sum f_iM_i^2-n\mu^2}{n-1}”

where f_{i} is frequency of i-th range,

M_{i} is midpoint of i-th range.

“\sigma^2=\frac{4\cdot14.5^2+8\cdot24.5^2+12\cdot34.5^2+22\cdot44.5^2+48\cdot54.5^2+6\cdot64.5^2-100\cdot46.5^2}{99}=\frac{231025-216225}{99}=149.49”

standard deviation:

“\sigma=\sqrt{149.49}=12.23”

iv)

32% percentile:

“P_{32}=l+w(32-c)/f”

where l is lower boundary of the class containing P_{32},

w is class interval size of the class containing P_{32},

f is frequency of the class containing *P*_{32},

c is cumulative frequency of the class preceding the class containing *P*_{32}

class containing P_{32}: 40-49

Then:

“P_{32}=40+9(32-24)/22=43.27”

v)

“mode=l+\frac{f_m-f_1}{(f_m-f_1)(f_m-f_)}\cdot h”

where l is lower limit of the modal class,

f_{m} is frequency of the modal class,

f_{1} is frequency of class preceding the modal class,

f_{2} is frequency of class succeeding the modal class,

h is width of the modal class.

modal class (class that has the highest frequency): 50-59

“mode=50+\frac{48-22}{48-22+48-6}\cdot9=53.44”

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