This document contains information about different statistical concepts including sampling methods, inferences about population means, time series analysis, and index numbers. It discusses simple random sampling, systematic sampling, stratified sampling, and cluster sampling. It also covers estimating population parameters using confidence intervals and the standard error. The document provides examples of calculating confidence intervals for population means when the population standard deviation is known and unknown. It defines different components of a time series including trend, cycle, season, and irregular components. Finally, it outlines various index number formulas including unweighted, weighted by quantity, and weighted by price indexes.

1. Statistics
Statistics
Furkatov Kamronbek

2. 01
Sampling methods
04
Time series 2
02
Inferences about
population mean
05
Index numbers
03
Time series

3. Sampling methods
01
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4. What is sample?
In statistics sampling is the selection
of a subset (a statistical sample) of
individuals from within a statistical
population to estimate characteristics
of the whole population.

5. 1 Simple random sample –
randomly choosing elements
Activities
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Systematic sample – has a system
in choosing (every k-th element)
Stratisfied sample – dividing into
groups (stratas) and randomly selecting
elements
Cluster sample –
dividing into clusters
randomly choosing these
clusters

6. Central Limit Theorem
If a sample size is equal
or more than 30 the
sample mean will have
a normal distribution

7. Independ work
A drinks company produces 1200 bottles of pop every30 minutes.
For quality control purposes, 12 bottles are selected and checked.
Each bottle passes through the machine in a single file. Using a
systematic sampling technique, determine the bottles that will be
selected for the sample.

8. Solution
This is systematic sampling
1.Order the population and give each data entry a unique reference number.
As each bottle passes through the machine in a single file, we can assume that the
first bottle has a reference number 1, the second number 2, etc.
We calculate the number of items of data in the sample
As we want a sample of 1200 bottles and we are using a systematic sample, we need to
choose the bottles using a sequence.
And we need divide population size into sampling size
So we need to pick every 100th term in the data.
3.Use a random number generator to select the first item of data.
As we need to pick every 100th , the first number that will form the starting point in the sample
selection must be randomly chosen from the first 100 terms. Using a random number
generator, we get the number 27, so we choose the first item of data in the sample to be
the 27th bottle.

9. As we are selecting every 100th item in regular
intervals, the next bottle will be number 127,
227, 327,127,227,327, and so on until we
reach the 1200th bottle in 30 minutes.

10. 02
INFERENCES
ABOUT
POPULATION
MEAN

11. Estimation of parameter
𝑥 ± 𝑧𝑎/2 ∗
𝜎
𝑛
𝑥 ± 𝑧𝑎/2 ∗
s
𝑛
100(1a)% Confidence Interval for 𝛍, 𝛔 known.
100(1a)% Confidence Interval for 𝛍, 𝛔 unknown.
We need to know these to answer the following
questions
“What is the average age in the class”?
“How much time do people spend reading books a day?”
“What is the average cost to spend a month in Tashkent?”

12. Standart error
● The standard error of the sample mean (x) is the sample standard deviation, and
shows how far the sample mean will be from the population mean (𝜇), on
average, in repeated random samples of size n.
● T procedures are very similar to z procedures, and they are used when the data
are not perfectly Normal and when the population standard deviation is
unknown. T procedures use the standard deviation of the sample instead of the
standard deviation of the population. The notation changes from sigma to s
when t procedures are used
When all of these assumptions are met, z scores can be used in the computation
process. However, many times these assumptions are not met and even more often
the population standard deviation is not known for the variable of interest. In this
case, t procedures are used instead, which are based on a distribution of
standardized scores called t scores.

13. ● After estimating the standard error, researchers can compute confidence
intervals and conduct tests of significance. Again, the same formula is used as
with the z procedures, except the sample standard deviation is used instead of
the population standard deviation. The other difference is that the notation t is
now used instead of z* in the confidence interval formula, and a t test statistic
instead of the z test statistic for the tests of significance

14. Independ work
m = 1,8 ± 2,575 *
0,622
10
=
In order to determine how many hours students spend
reading books in a day, a survey was conducted among
n=10 students. s=0.622, x*=1.8 h
Solution:
x*=1.8; n=10 s=0,622
99% > 2,575 1,294 ≤ 𝜇 ≤ 2,306
We’re sure in 99% average spent time by students for reading books is between 1,294
hour and 2,306 hour

15. TIME SERIES
03

16. Time series
Trend
Cycle
Season
Irregular
Types of components
a =
Σ𝑦
𝑛
𝑦𝑡 = 𝑎 + 𝑏𝑡
𝑏 =
Σ𝑦𝑡
Σ𝑡2
Fitted value function

17. Time series of Artel company

18. Index numbers
04
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19. Indexes
Unweighted value
Unweighted quantity
Weighted value
Weighted price
𝐼𝑝𝑞
𝑖𝑝𝑞
𝐼𝑞
𝑖𝑞
𝐼𝑝
𝑖𝑃
Unweighted indexes
Weighted quantity
Weighted indexes
Unweighted price

20. Formulas
Unw. price
𝑖𝑝 =
𝑝1
𝑝0
𝑖𝑞 =
𝑞1
𝑞0
𝑖𝑝𝑞 =
𝑝1𝑞1
𝑝0𝑞0
Unw. quantity
Unw. value
𝑝1; 𝑞1 − 𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑝0; 𝑞0 −𝑏𝑎𝑠𝑖𝑠
W. price
Paasche’s
𝐼𝑝 =
Σ𝑝1𝑞1
Σ𝑝0𝑞1
𝐼𝑞 =
Σ𝑝0𝑞1
Σ𝑝0𝑞0
𝐼𝑝𝑞 =
Σ𝑝1𝑞1
Σ𝑝0𝑞0
W. quantity
Laspeyre’s
W. value

21. Example (independent work)
OCT NOV OCT NOV
q0 p0 q1 p1
Child. 10 1000000 14 1050000
Kitchen 14 2500000 15 2450000
Soft 13 1200000 18 1280000
Unw. P Unw. Q Unw. V
Child. 1,05 1,4 1,47
Kitchen 0,98 1,071429 1,05
Soft 1,066667 1,384615 1,476923
W.P 1,019015
W.Q 1,206271
W.V 1,229208
The value of A company increased by 23%, because of quantity +2% and price +21%