Answer to Question #182003 in Statistics and Probability for EUGINE HAWEZA

Question

(1) Dr Mukonda’s performance of his 170 Biostatistics class of students is given in an incomplete distribution below.

Variable 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Frequency f1 21 32 40 21 26 f7

(a) If the median is 35. After finding the missing frequencies, find the standard deviation and present this information

on a pie chart.

(b) Statistically, explain if these results are normally distributed.

Answer 1

Class interval Frequency Cumulative Frequency 

 0−10 10 10 

 10−20 20 30 

 20−30 $f_1$ 30+$f _1$

 30−40 40 70+$f_1$

 40−50 $f_2$  70+$f _1+f_ 2$

​ 50−60 25 95+$f_1+f_2$

 60−70 15 110+$f _1+f_2$

​

  

(a) Here, Median=35.

∴ median class is 30−40,N=170,then $\dfrac{N}{2}=\dfrac{170}{2}=85$

$L=30,f=40,F=30+f_ 1,h=10$

⇒ Now,  Median$=L+\dfrac{\dfrac{N}{2}-F}{f}\times h$

$35=30+\dfrac{85-(30+f_1)}{40}\times 10$

 $\Rightarrow 35\times 4=120+(55-f_1)$

$\Rightarrow f_1=35$

 $Also f_2=170-110-f_1=60-35=25$

​

Hence, the missing frequencies are,

Class 20−30⇒35

Class 40−50⇒25

The table for the above data distribution is-

Mean $\mu=\dfrac{\sum XF}{\sum F}=\dfrac{6100}{170}=35.88$

Standard deviation $s=\sqrt{\dfrac{\sum (x-\mu)^2}{N}}=\sqrt{\dfrac{3016.59}{170}}=\sqrt{17.25}=4.212$

The pie chart is-

(b) The given data is not normally distributed since, There is no decrement in the frequencies after the middle frequency.