Answer to Question #194657 in Statistics and Probability for ndinoo

Question #194657

The following table is a frequency table of the scores obtained in a competition. Use

the table answer the questions below.

Classes Frequency(f)10 - 13 413 - 16 616 - 19 1219 - 22 1422 - 25 4Total 40a. Find the mean, median and mode of the score. [2,2,2]

b. Find the range, variance, and standard deviation. [1,3,1]

c. Find the coefficient of variation. [2]

d. Compute the interquartile range.


1
Expert's answer
2022-01-10T19:01:15-0500

a.


Classfmdfdfd2cf1013411.5281641316614.51661016191217.50002219221420.511414362225423.5281640\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} Class & f & m & d & f\cdot d & f\cdot d^2 & cf\\ \hline 10-13 & 4 & 11.5 &-2 &- 8 & 16 & 4 \\ 13-16 & 6 & 14.5 & -1 & -6 & 6 & 10 \\ 16-19 & 12 & 17.5 & 0 & 0 & 0 & 22 \\ 19-22 & 14 & 20.5 & 1& 14 & 14 & 36 \\ 22-25 & 4 & 23.5 & 2 & 8 & 16 & 40 \\ \end{array}



A=17.5A=17.5

d=xAh,h=3d=\dfrac{x-A}{h}, h=3



Mean=xˉ=A+fdnhMean=\bar{x}=A+\dfrac{\sum f d}{n}\cdot h=17.5+8403=18.1=17.5+\dfrac{8}{40}\cdot 3=18.1

The median class is 16-19.

L=16,n=40L=16, n=40

Cumulative frequency of the class preceding the median class cf=10cf=10

Frequency of the median class f=12f=12

Class length of median class c=3c=3

Median M=L+n2cffcMedian\ M=L+\dfrac{\dfrac{n}{2}- cf}{f}\cdot c=16+40210123=18.5=16+\dfrac{\dfrac{40}{2}- 10}{12}\cdot 3=18.5


Maximum frequency is14.

The mode class is 19-22.

L=19L=19

Cumulative frequency of the class preceding the median class cf=10cf=10

Frequency of the mode class f1=14f_1=14

Frequency of the preceding class f0=12f_0=12

Frequency of the succeeding class f2=4f_2=4

Class length of mode class c=3c=3



Z=L+f1f02f1f0f2cZ=L+\dfrac{f_1-f_0}{2f_1-f_0-f_2}\cdot c=19+14122(14)1243=19.5=19+\dfrac{14-12}{2(14)-12-4}\cdot 3=19.5

b.



Range=22+25210+132=12Range=\dfrac{22+25}{2}-\dfrac{10+13}{2}=12S2=fd2(fd)2nn1h2S^2=\dfrac{\sum f\cdot d^2-\dfrac{(\sum f\cdot d)^2}{n}}{n-1}\cdot h^2=52(8)24040132=151.21311.63077=\dfrac{52-\dfrac{(8)^2}{40}}{40-1}\cdot 3^2=\dfrac{151.2}{13}\approx11.63077S=S2=151.2133.4104S=\sqrt{S^2}=\sqrt{\dfrac{151.2}{13}}\approx3.4104

c.


Coefficient of Variation=Sxˉ100%Coefficient \ of \ Variation=\dfrac{S}{\bar{x}}\cdot 100\%=151.21318.1100%18.842%=\dfrac{\sqrt{\dfrac{151.2}{13}}}{18.1}\cdot 100\%\approx18.842\%

d.

Class with (n4)th(\dfrac{n}{4})^{th}  value of the observation in cfcf column

=(404)th=(\dfrac{40}{4})^{th} value of the observation in cfcf column

=(10)th=(10)^{th} value of the observation in cfcf column

and it lies in the class 13-16.

Q1Q_1 class: 131613-16

L=13L=13



Q1=L+n4cffcQ_1=L+\dfrac{\dfrac{n}{4}- cf}{f}\cdot c=13+404463=16=13+\dfrac{\dfrac{40}{4}- 4}{6}\cdot 3=16

Class with (3n4)th(\dfrac{3n}{4})^{th}  value of the observation in cfcf column

=(3404)th=(\dfrac{3\cdot 40}{4})^{th} value of the observation in cfcf column

=(30)th=(30)^{th} value of the observation in cfcf column

and it lies in the class 19-22.

Q3Q_3 class: 192219-22

L=19L=19



Q3=L+3n4cffcQ_3=L+\dfrac{\dfrac{3n}{4}- cf}{f}\cdot c=19+34042214320.7143=19+\dfrac{\dfrac{3\cdot 40}{4}- 22}{14}\cdot 3\approx20.7143IQR=Q3Q120.7143164.7143IQR=Q_3-Q_1\approx20.7143-16\approx4.7143




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