Question

12. Consider the following data set of a given characteristic. 34.35242 28-62 86.66 51-29 2020 14.32 39-08 2 14:07 | 11-03 | 2866 | 3370 | 90HT 2675 188.29 28:61 26-26 31-03 (a) Construct a frequeney table with four (4) classes. (b) Draw a histogram. (c) Find the median of the data, the first quartile, and the third quartile. 52.5 (d) Find the population variance and the population standard devia- 32.115 12.1S tion. V (e) Find the sample variance and the sample standard deviation. (f) Determine the population z-score corresponding to the score 67.18. ean ヴ (g) Construct a 90% confidence interval for the mean score having this given characteristic.

Answer 1

(a)

C.I	frequency
13.07-37.07	19
37.07-61.07	6
61.07-85.07	3
85.07-109.07	2
Total	30

(b)

histogram 20 18 16 14 12 10 13.07,37.07] 37.07, 61.07) (85.07, 109.07 (61.07,85.07)

(c) N=30 , N/2 =15

C.I	frequency	cf
13.07-37.07	19	19
37.07-61.07	6	25
61.07-85.07	3	28
85.07-109.07	2	30
Total	30

The value just greater than 15 in cf column is 19 , corresponding CI is 13.07-37.07

13.07-37.07 is the median class

Median = L + h* (N/2 -c)/f

where L = lower limit if median class = 13.07

h = width =24

c = cumulative frequency of the class preceding the median class =0

f = frequency of the median class =19

Therefore

Median = 32.02

Q1= L + h* (N/4 -c)/f

L=13.07, h =24, c =0, f =19

Q1=22.54

Q3= L + h* (3N/4 -c)/f

L= 37.07, h=24, c=19, f= 6

Q3= 51.07

(d)

C.I	frequency	x	fx	fx^2
13.07-37.07	19	25.07	476.33	11941.59
37.07-61.07	6	49.07	294.42	14447.19
61.07-85.07	3	73.07	219.21	16017.67
85.07-109.07	2	97.07	194.14	18845.17
	30		1184.1	61251.63

$\sigma ^{2}=\sum f.x^{2}/N- (\sum fx/N)^{2}$

= 483.84

$\sigma = \sqrt{\sum f.x^{2}/N- (\sum fx/N)^{2}}$

= 21.9964

(e)

$s^{2}=\sum f.x^{2}/(N-1)- (\sum fx)^{2}/N(N-1)$

= 500.5241

s= 22.3724

(f) z score =

SCOTemiedm escoTE

= (67.18-39.47) / 21.9964

= 1.26

Note :

Mean=〉 '.fr/N = 39.47

(g) 90% Confidence interval is

$\bar{X}\pm z_{c}*\sigma /\sqrt{n}$

=

39.47 ± 1.645 * 21.9964/v/30

= (32.86, 46.07)

Note : For 90% confidence , zc =1.645