Answer to Question #91039 in Statistics and Probability for ABU

Question #91039
The following table shows the levels of retirement benefits given to a group of
workers in a given establishment.

Retirement benefits £ ‘000 No of retirees (f)
20 – 29 50
30 – 39 69
40 – 49 70

Required:-
Calculate the percentile coefficient of Kurtosis and briefly comment on the value
obtained
50 – 59 90
60 – 69 52
70 – 79 40
80 – 89 11
1
Expert's answer
2019-06-27T12:48:59-0400
Class IntervalsFrequencies (fi)Cumulative Frequencies20295050303969119404970189505990279606952331707940371808911382\def\arraystretch{1.5} \begin{array}{c:c:c} Class \ Intervals & Frequencies\ (f_i) & Cumulative\ Frequencies \\ \hline 20-29 & 50 & 50 \\ 30-39 & 69 & 119 \\ 40-49 & 70 & 189 \\ 50-59 & 90 & 279 \\ 60-69 & 52 & 331 \\ 70-79 & 40 & 371 \\ 80-89 & 11 & 382 \\ \end{array}fi=382\sum f_i=382

The percentile coefficient of Kurtosis, denoted as Kp,K_p, is defined in terms of quartiles and percentiles as


Kp=Q3Q12(P90P10)K_p={Q_3-Q_1 \over 2(P_{90}-P_{10})}

Q1=L1+[fi×1/4Cff1]C1Q_1=L_1+\bigg[{\sum f_i\times 1/4-C_f \over f_1}\bigg]\cdot C_1

fi×1/4=382×1/4=95.5,L1=30,\sum f_i\times 1/4=382\times 1/4=95.5, L_1=30,Cf=50,C1=10,f1=69C_f=50, C_1=10, f_1=69

Q1=30+[95.55069]10=36.5942Q_1=30+\bigg[{95.5-50 \over 69}\bigg]\cdot 10=36.5942

Q3=L3+[fi×3/4Cff3]C3Q_3=L_3+\bigg[{\sum f_i\times 3/4-C_f \over f_3}\bigg]\cdot C_3

fi×3/4=382×3/4=286.5,L3=60,\sum f_i\times 3/4=382\times 3/4=286.5, L_3=60,Cf=279,C3=10,f3=52C_f=279, C_3=10, f_3=52

Q3=60+[286.527952]10=61.4423Q_3=60+\bigg[{286.5-279 \over 52}\bigg]\cdot 10=61.4423

Pk=L1+[fi×(k/100)Cffk]CkP_k=L_1+\bigg[{\sum f_i\times (k/100)-C_f \over f_k}\bigg]\cdot C_k

k=90,fi×(k/100)=382×(90/100)=343.8,k=90, \sum f_i\times (k/100)=382\times (90/100)=343.8,L1=70,Cf=331,C90=10,f90=40L_1=70, C_f=331, C_{90}=10, f_{90}=40

P90=70+[343.833140]10=73.20P_{90}=70+\bigg[{343.8-331 \over 40}\bigg]\cdot 10=73.20

k=10,fi×(k/100)=382×(10/100)=38.2,k=10, \sum f_i\times (k/100)=382\times (10/100)=38.2,L1=20,Cf=0,C10=10,f10=50L_1=20, C_f=0, C_{10}=10, f_{10}=50


P10=10+[38.2050]10=17.64P_{10}=10+\bigg[{38.2-0 \over 50}\bigg]\cdot 10=17.64

The percentile coefficient of Kurtosis


Kp=61.442336.59422(73.2017.64)=0.2236K_p={61.4423-36.5942 \over 2(73.20-17.64)}=0.2236

Since Kp=0.2236<0.263K_p=0.2236<0.263, the curve is leptokurtic or thin.



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