STA 304H1F Winter 2020 Assignment 2: Analysis of Sampling Techniques

This assignment focuses on various sampling techniques, including stratified and systematic sampling, and their application in statistical analysis. The first question explores stratified sampling with a small population, calculating means, variances, and possible sample combinations. The second question addresses sample size determination for estimating a population proportion across different strata with varying characteristics. The third question involves using R to analyze sales data from two brands, using the provided data, and the fourth question uses R to analyze a dataset of student marks, constructing histograms, calculating means and standard deviations, and comparing the effectiveness of single and repeated systematic sampling methods, including the generation of confidence intervals. The assignment requires the use of R for data analysis, interpretation of results, and the construction of confidence intervals, demonstrating an understanding of statistical concepts and their practical application.

Running Head: ASSIGNMENT ON SAMPLING TECHNIQUES
ASSIGNMENT ON SAMPLING TECHNIQUES
Name of the Student:
Name of the University:
Author Note:

1ASSIGNMENT ON SAMPLING TECHNIQUES
Answer 1
Here a population of 6 students is divided into two equal sized strata. It is given that,
N=6 , N1 =3 , N 2=3 , n1=2 , n2=2
Where N=total population size, Ni=size of the ith strata, ni=sample size taken from the ith strata.
a) μ1= 1
3 [66+59+ 70 ] =65
μ2= 1
3 [83+82+ 71 ] =78.67
b) σ 2= 1
N ∑
i=1
N
( Y i−Y ) 2 , Y i’s are the observations.
σ 1
2=1
3 [ ( 66−65 )2 + ( 59−65 )2+ (70−65 )2 ] =20.67
σ 2
2=1
3 [ ( 83−78.67 )2 + ( 82−78.67 )2+ ( 71−78.67 )2 ]=29.56
c) S2= 1
N−1 ∑
i=1
N
( Y i−Y ) 2 ⇒ S1
2 =31, S2
2=44.33
Var ( ^τ1 ) =N1
2
( N 1−n1
N1 n1 )S1
2=46.5,τi is the population total.
Var ( ^τ2 ) =N2
2
( N 2−n2
N 2 n2 )S2
2=66.50
d) Suppose a stratified sample is constructed taking scores of 1st and 2nd students (66,59)
from stratum 1 and scores of 4th and 5th students (83,82) from stratum 2.
n1=2, n2 =2 , y1=62.5 , y1=82.5
Var ( ^μstr )= 1
N2 ∑ Ni ( N i−ni ) Si
2
ni
= 1
36 [ 3. 31
2 +3. 44.33
2 ] =3.14

2ASSIGNMENT ON SAMPLING TECHNIQUES
Var ( ^τ str ) =N2 Var ( ^μstr ) =113.04
e) If SRSWOR(Simple Random Sampling Without Replacement) is adopted to form a
stratified sample of size 4, taking 2 from each strata, then the number of all possible cases
is C2
3 ∗ C2
3 =9
The samples are-
((66, 59),(83,82)), ((66,59),(83,71)), ((66,59),(82,71)), ((66,70),(83,82)), ((66,70),
(83,71)), ((66,70),(82,71)), ((59,70),(83,82)), ((59,70),(83,71)), ((59,70),(82,71)).
f)
g) μ= 1
N ∑
i=1
N
Y i=71.83
E ( ^μstr ) = 1
N ∑
i=1
N
Ni Y i =71.83=μ
Therefore ^μ is an unbiased estimator of μ. Similarly, ^τ is an unbiased estimator of τ .

3ASSIGNMENT ON SAMPLING TECHNIQUES
Answer 4
Here the dataset contains scores of 312 students. The necessary calculations are provided in a
R Markdown File.
a) The histogram shows that the distribution of test 1 scores is skewed to the left. The mean
and standard deviation are 70.79167=71(approx.) and 15.49539 respectively.
b) Here a systematic sample of size 20 is drawn from the population. The estimated mean
score is 65.8. The bound error estimate is found to be 7.155477. Hence, a 95%
confidence interval of the estimate is (58.64452, 72.95548).
c) Here 5 repeated systematic sample are drawn from the population. The estimated average
marks is found to be 67.82. The error bound of the estimate is equal to 9.565965. Hence,
the 95% confidence interval of the estimate is (58.25404, 77.38596).
d) The repeated sampling is better technique since it reduces the bias in sampling and gives
a wider confidence interval. Hence method in c is more appropriate in this case.

4ASSIGNMENT ON SAMPLING TECHNIQUES