Question

There are 180 primary schools in a country area having an average of 30 or more people under the age of 21 per class. A sample of 30 schools drawn using systematic sampling with an interval of k = 6.

Serial number	1	2	3	4	5	6	7	8	9	10
No. of students	60	200	45	50	40	79	35	41	30	120
Serial number	11	12	13	14	15	16	17	18	19	20
No. of students	300	65	111	120	200	42	51	67	32	40
Serial number	21	22	23	24	25	26	27	28	29	30
No. of students	46	55	250	100	63	90	47	82	31	50

1. Estimate total number of students.
2. Estimate average number of students per farm.
3. The variance of the sample mean of students per farm.
4. 95% confidence interval for the total number of students.

Answer 1

Serial number	x	( x - x¯)	( x - x¯)²
1	60	-24.73	611.5729
2	200	115.27	13287.1729
3	45	-39.73	1578.4729
4	50	-34.73	1206.1729
5	40	-44.73	2000.7729
6	79	-5.73	32.8329
7	35	-49.73	2473.0729
8	41	-43.73	1912.3129
9	30	-54.73	2995.3729
10	120	35.27	1243.9729
11	300	215.27	46341.1729
12	65	-19.73	389.2729
13	111	26.27	690.1129
14	120	35.27	1243.9729
15	200	115.27	13287.1729
16	42	-42.73	1825.8529
17	51	-33.73	1137.7129
18	67	-17.73	314.3529
19	32	-52.73	2780.4529
20	40	-44.73	2000.7729
21	46	-38.73	1500.0129
22	55	-29.73	883.8729
23	250	165.27	27314.1729
24	100	15.27	233.1729
25	63	-21.73	472.1929
26	90	5.27	27.7729
27	47	-37.73	1423.5529
28	82	-2.73	7.4529
29	31	-53.73	2886.9129
30	50	-34.73	1206.1729
	∑x = 2542		∑( x - x¯)² = 133307.867

The total number of students = 2542

Mean = ∑x / n

= 2542 / 25

x¯ = 84.73

Variance = ∑( x - x¯)² / ( n-1)

  = 133307.867 / 29

Variance = 4596.823  

Standard Deviation = √4596.823 = 67.80

Sample size n = 30

Sample Standard deviation S = 67.80

mean μ = 84.73

Standard Error = S / √n

                                          = 67.80 / √30

                                          = 12.37

               Standard Error = 12.37

Confidence Interval = Mean + - Margin Error

Confidence Interval = Mean + - ( Critical value* Standard Error)

For Critical value:

               ꭤ = 1 – 0.95 = 0.05

P value = 1 – (ꭤ/ 2)

              = 1 – (0.05/2)

              = 1 – 0.025

              = 0.975

From Z table we get

Critical value = 1.96

Now,

Confidence Interval = Mean + - ( Critical value* Standard Error)

                                      = 84.73 + - ( 1.96 * 12.37)

                                      = 84.73 + - 24.24

                                                     = 60 and 109

Confidence Interval = 60 and 109