Statistical Inference

• Faculty of Economics •

Gazdaságelméleti és Módszertani Intézet

Statistical Inference 1st lecture

Petra Petrovics



Statistics Descriptive - it is concerned only with collecting and describing data

Population

Inferential - it is used when tentative conclusions about a population are drawn on the basis of a sample

Sample

- the portion of the population

- set of elements - set of all possible measurements - the number of elements: N or 

- about which information is gathered - representative - the number of elements: n - „simple random” sample



Basic Terms I • Parameter (Θ) → a characteristic of a population → e.g. average, proportion, variance

• Statistic → a characteristic of a sample → e.g. average, proportion, variance

• Representative sample The values in the sample must be typical of values in the population

• Random sample Any sample of size n has the same chance (probability) of being selected


• Statistical error


Basic Terms II

1. Non-sampling error • harder to quantify • Systematic error Processing error Not appropriate supplying of data, etc.

2. Sampling error • Using sample rather than population • It can be quantified • Depends on: → Population distribution → Sampling method → Sample size → Statistical method



Inference from the Sample to the Population

Estimation

Hypothesis Testing

Estimation: how can we determine the value of an unknown parameter of a population by using the sample. Hypothesis Testing: how to test a statement concerning a population parameter.



Estimator A tool for statistical inference; sample statistics are used to estimate population parameters.

General estimator criteria 1. 2. 3. 4. 5.

Estimation costs Goodness of fit Unbiased Efficiency Asymptotic characteristics



1. Estimation cost 2. Goodness of fit Model estimation vs. observed data

3. Unbiased If its expected value is equal to the population parameter it estimates. Any systematic deviation of the estimator away from the parameter of interest is called bias. Unbiased: the bias is zero.



Unbiased Estimator

Θ is unbiased

biased, because E(Θ)≠0



Example for Unbiased Estimator • The following are data about the salary of employees of a small enterprise (th HUF): 180, 90, 36, 30 • Estimate the average salary using the: – Sample mean; – Median; – Central point (the average of the minimum and maximum value) as an estimator.



Characteristics of Samples Seq. number 1st 2nd 3rd

Elements

Mean

Median

Central Point

30, 36, 90 30, 36, 180 30, 90, 180

52 82 100

36 36 90

60 105 105

36, 90, 180 Expected value

102 84

90 63

108 94.5

4th



4. Efficiency if it has a relatively small variance (and standard deviation)



5. Asymptotic characteristics a) Consistency If its probability of being close to the parameter it estimates increases as the sample size increases



5. Asymptotic characteristics b) Asymptotic normality n grows  approaches a normal distribution Central Limit Theorem: the distribution of independent observations tend to close to the normal distribution, if the sample size is enough large. Large samle: – n ≥ 100 – In case of unimodal distribution: n ≥ 30



Estimation Classical Least Squares Maximum Method Likelihood

Bayesian Robostness

1. Point estimators 2. Interval estimators



Basic Terms I • Point Estimate: the value of estimator; a single number that is used to estimate an unknown parameter • Confidence Level: specific percentage π • Confidence Interval (CI): an interval estimate is a range of values used to estimate a population parameter   P l    u   π or Θ → ± ΔΘ



Number of samples

Parameters and Confidence Intervals

Heights (cm)

Results: • Changes from sample to sample • Are around the statistical parameter • n↑ , standard deviation ↓



Basic Terms II Θ → ± ΔΘ

• Maximum Error: Δ = z π or t   standard error • Standard Error: standard deviation of the estimators • zπ : when test statistics are approximately normally distributed for large samples; n  100 • tπ : Student's t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small; n < 100



Basic Terms III • Degrees of freedom (df) – The number of values in the final calculation of a statistic that are free to vary. – The number of independent pieces of information that go into the estimate of a parameter. – In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate (n) minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself.

x

(x)

x

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48 0,50

0,5000 0,5080 0,5160 0,5239 0,5319 0,5398 0,5478 0,5557 0,5636 0,5714 0,5793 0,5871 0,5948 0,6026 0,6103 0,6179 0,6255 0,6331 0,6406 0,6480 0,6554 0,6628 0,6700 0,6772 0,6844 0,6915

0,52 0,54 0,56 0,58 0,60 0,62 0,64 0,66 0,68 0,70 0,72 0,74 0,76 0,78 0,80 0,82 0,84 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00 1,02

(x)

x x (x) (x) • Faculty of Economics

0,6985 • 0,7054 0,7123 0,7190 0,7257 0,7324 0,7389 0,7454 0,7517 0,7580 0,7642 0,7703 0,7764 0,7823 0,7881 0,7939 0,7995 0,8051 0,8106 0,8159 0,8212 0,8264 0,8315 0,8365 0,8413 0,8461

1,04 1,06 1,08 1,10 1,12 1,14 1,16 1,18 1,20 1,22 1,24 1,26 1,28 1,30 1,32 1,34 1,36 1,38 1,40 1,42 1,44 1,46 1,48 1,50 1,52 1,54

0,8508 0,8554 0,8599 0,8643 0,8686 0,8729 0,8770 0,8810 0,8849 0,8888 0,8925 0,8962 0,8997 0,9032 0,9066 0,9099 0,9131 0,9162 0,9192 0,9222 0,9251 0,9279 0,9306 0,9332 0,9357 0,9382

1,56 1,58 1,60 1,62 1,64 1,66 1,68 1,70 1,72 1,74 1,76 1,78 1,80 1,82 1,84 1,86 1,88 1,90 1,92 1,94 1,96 1,98 2,00 2,10 2,20 2,30

0,9406 0,9429 0,9452 0,9474 0,9495 0,9515 0,9535 0,9554 0,9572 0,9591 0,9608 0,9625 0,9641 0,9656 0,9671 0,9686 0,9699 0,9713 0,9726 0,9748 0,9750 0,9761 0,9772 0,9821 0,9861 0,9893

x

(x)

2,40 2,50 2,60 2,70 2,80 2,90 3,00 3,20 3,40 3,60 3,8

0,9918 0,9938 0,9953 0,9965 0,9974 0,9981 0,9987 0,9993 0,9996 0,9998 0,9999


z-test ( x) 

1  2

Df 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 

t-test • Student’s Faculty of Economics 0,55

0,60

0,70

0,158 0,142 0,137 0,134 0,132 0,131 0,130 0,130 0,129 0,129 0,129 0,128 0,128 0,128 0,128 0,128 0,128 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,127 0,126 0,126 0,126 0,126

0,325 0,289 0,277 0,271 0,267 0,265 0,263 0,262 0,261 0,260 0,260 0,259 0,259 0,258 0,258 0,258 0,257 0,257 0,257 0,257 0,257 0,256 0,256 0,256 0,256 0,256 0,256 0,256 0,256 0,256 0,255 0,254 0,254 0,253

0,727 0,617 0,584 0,569 0,559 0,553 0,549 0,546 0,543 0,542 0,540 0,539 0,538 0,537 0,536 0,535 0,534 0,534 0,533 0,533 0,532 0,532 0,532 0,531 0,531 0,531 0,531 0,530 0,530 0,530 0,529 0,527 0,526 0,524

0,75

0,80

0,90

0,95

0,975

0,99

0,995

6,96 4,54 3,75 3,36 3,14 3,00 2,90 2,82 2,76 2,72 2,68 2,65 2,62 2,60 2,58 2,57 2,55 2,54 2,53 2,52 2,51 2,50 2,49 2,48 2,48 2,47 2,47 2,46 2,46 2,42 2,39 2,36 2,33

63,66 9,92 5,84 4,60 4,03 3,71 3,50 3,36 3,25 3,17 3,11 3,06 3,01 2,98 2,95 2,92 2,90 2,88 2,86 2,84 2,83 2,82 2,81 2,80 2,79 2,78 2,77 2,76 2,76 2,75 2,70 2,66 2,62 2,58

• Gazdaságelméleti és Módszertani Intézet 1,000 1,376 3,08 6,31 12,71 31,82 0,816 0,765 0,741 0,727 0,718 0,711 0,706 0,703 0,700 0,697 0,695 0,694 0,692 0,691 0,690 0,689 0,688 0,688 0,687 0,686 0,686 0,685 0,685 0,684 0,684 0,684 0,683 0,683 0,683 0,681 0,679 0,677 0,674

1,061 0,978 0,941 0,920 0,906 0,896 0,889 0,883 0,879 0,876 0,873 0,870 0,868 0,866 0,865 0,863 0,862 0,861 0,860 0,859 0,858 0,858 0,857 0,856 0,856 0,855 0,855 0,854 0,854 0,851 0,848 0,845 0,842

1,89 1,64 1,53 1,48 1,44 1,42 1,40 1,38 1,37 1,36 1,36 1,35 1,34 1,34 1,34 1,33 1,33 1,33 1,32 1,32 1,32 1,32 1,32 1,32 1,32 1,31 1,31 1,31 1,31 1,30 1,30 1,29 1,28

2,92 2,35 2,13 2,02 1,94 1,90 1,86 1,83 1,81 1,80 1,78 1,77 1,76 1,75 1,75 1,74 1,73 1,73 1,72 1,72 1,72 1,71 1,71 1,71 1,71 1,70 1,70 1,70 1,70 1,68 1,67 1,66 1,645

4,30 3,18 2,78 2,57 2,45 2,36 2,31 2,26 2,23 2,20 2,18 2,16 2,14 2,13 2,12 2,11 2,10 2,09 2,09 2,08 2,07 2,07 2,06 2,06 2,06 2,05 2,05 2,04 2,04 2,02 2,00 1,98 1,96



To estimate… 1. Select a random sample from the population of interest. 2. Calculate the point estimate of the parameter. 3. Calculate a measure of its variability, often a confidence interval (CI). 4. Associate with this estimate a measure of variability.



Estimation elements mean standard deviation proportion

Population X1, X2, … , XN, … μ Σ

Sample x1, x2, … , xn

P

p

x s



A.) μ – mean or expected value of the population

μ  x  x 1.) normal population, σ known μ  x  zπ  σ x

σx 

 n

 1-

If n N

n  10% N

2.) normal population, σ unknown, n  100 μ  x  zπ  s x

3.) normal population, σ unknown, n < 100 μ  x  tπ  s x

sx 

s n  1N n


B)


P – the proportion of the population

= population proportion is equal to the number of elements in the population belonging to the category of interest, divided by the total number of elements in the population

P

 p  p

 p  z  s p  z 

In case of large sample, when n ≥ 100!

p  1  p  n


C)


σ – the standard deviation of the population  2 2   n - 1  s n - 1  s  2  P  σ   π 2  χ2  χα 2   1 - α 2  

Only in the case when the population distribution is normal!

χ • 0,25 Faculty of0,75Economics 0,50 0,90 0,95 2

Df 1

0,005 0,0000

0,01 0,0002

0,025 0,0010

0,05 0,039

0,10 0,0158

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

0,0100 0,072 0,207 0,412 0,676 0,989 1,34 1,73 2,16 2,60 3,07 3,57 4,07 4,60 5,14 5,70 6,26 6,84 7,43 8,03 8,64 9,26 9,89 10,5 11,2 11,8 12,5 13,1 13,8 20,7 28,0 35,5 43,3 51,2 59,2 67,3

0,0201 0,115 0,297 0,554 0,872 1,24 1,65 2,09 2,56 3,05 3,57 4,11 4,66 5,23 5,81 6,41 7,01 7,63 8,26 8,90 9,54 10,2 10,9 11,5 12,2 12,9 13,6 14,3 15,0 22,2 29,7 37,5 45,4 53,5 61,8 70,1

0,0506 0,216 0,484 0,831 1,24 1,69 2,18 2,70 3,25 3,82 4,40 5,01 5,63 6,26 6,91 7,56 8,23 8,91 9,59 10,3 11,0 11,7 12,4 13,1 13,8 14,6 15,3 16,0 16,8 24,4 32,4 40,5 48,8 57,2 65,6 74,2

0,103 0,352 0,711 1,15 1,64 2,17 2,73 3,33 3,94 4,57 5,23 5,89 6,57 7,26 7,96 8,67 9,39 10,1 10,9 11,6 12,3 13,1 13,8 14,6 15,4 16,2 16,9 17,7 18,5 26,5 34,8 43,2 51,7 60,4 69,1 77,9

0,211 0,584 1,06 1,61 2,20 2,83 3,49 4,17 4,87 5,58 6,30 7,04 7,79 8,55 9,31 10,1 10,9 11,7 12,4 13,2 14,0 14,8 15,7 16,5 17,3 18,1 18,9 19,8 20,6 29,1 37,7 46,5 55,3 64,3 73,3 82,4

•

0,975 5,02

0,99 6,63

0,995 7,88

7,38 9,35 11,1 12,8 14,4 16,0 17,5 19,0 20,5 21,9 23,3 24,7 26,1 27,5 28,8 30,2 31,5 32,9 34,2 35,5 36,8 38,1 39,4 40,6 41,9 43,2 44,5 45,7 47,0 59,3 71,4 83,3 95,0 106,6 118,1 129,6

9,21 11,3 13,3 15,1 16,8 18,5 20,1 21,7 23,2 24,7 26,2 27,7 29,1 30,6 32,0 33,4 34,8 36,2 37,6 38,9 40,3 41,6 43,0 44,3 45,6 47,0 48,3 49,6 50,9 63,7 76,2 88,4 100,4 112,3 124,1 135,8

10,6 12,8 14,9 16,7 18,5 20,3 22,0 23,6 25,2 26,8 28,3 29,8 31,3 32,8 34,3 35,7 37,2 38,6 40,0 41,4 42,8 44,2 45,6 46,9 48,3 49,6 51,0 52,3 53,7 66,8 79,5 92,0 104,2 116,3 128,3 140,2

0,102 0,455 1,32 és 2,71 3,84 Gazdaságelméleti Módszertani Intézet 0,575 1,21 1,92 2,67 3,45 4,25 5,07 5,90 6,74 7,58 8,44 9,30 10,2 11,0 11,9 12,8 13,7 14,6 15,5 16,3 17,2 18,1 19,0 19,9 20,8 21,7 22,7 23,6 24,5 33,7 42,9 52,3 61,7 71,1 80,6 90,1

1,39 2,37 3,36 4,35 5,35 6,35 7,34 8,34 9,34 10,3 11,3 12,3 13,3 14,3 15,3 16,3 17,3 18,3 19,3 20,3 21,3 22,3 23,3 24,3 25,3 26,3 27,3 28,3 29,3 39,3 49,3 59,3 69,3 79,3 89,3 99,3

2,77 4,11 5,39 6,63 7,84 9,04 10,2 11,4 12,5 13,7 14,8 16,0 17,1 18,2 19,4 20,5 21,6 22,7 23,8 24,9 26,0 27,1 28,2 29,3 30,4 31,5 32,6 33,7 34,8 45,6 56,3 67,0 77,6 88,1 98,6 109,1

4,61 6,25 7,78 9,24 10,6 12,0 13,4 14,7 16,0 17,3 18,5 19,8 21,1 22,3 23,5 24,8 26,0 27,2 28,4 29,6 30,8 32,0 33,2 34,4 35,6 36,7 37,9 39,1 40,3 51,8 63,2 74,4 85,5 96,6 107,6 118,5

5,99 7,81 9,49 11,1 12,6 14,1 15,5 16,9 18,3 19,7 21,0 22,4 23,7 25,0 26,3 27,6 28,9 30,1 31,4 32,7 33,9 35,2 36,4 37,7 38,9 40,1 41,3 42,6 43,8 55,8 67,5 79,1 90,5 101,9 113,1 124,3



Example 1 • From the population of BA students a sample of 15 students was taken. • Confidence level: π = 95 % • A random sample (in days): 5, 8, 12, 4, 9, 11, 12, 14, 9, 7, 6, 11, 9, 8, 10 A) Estimate the average time spent the BA students on Statistics! B) Estimate the standard deviation of time spent on Statistics!



A) 1) - normal distribution - population standard deviation: 2 days μ  x  z 

 n

 9  1.96 

2 15

2) - population standard deviation is unknown - sample standard deviation: s = 2.7 s 2.7 μ  x  t   9  2.14  n 15

B) Standard deviation!

14  2.72  1.98  σ  26.1

14  2.72  4.26 5.63



Example 2 • The management of a manufacturer of calculators and microcomputers wants to improve the quality of their products. • 150-element sample was drawn from a lot of calculators. They tested each of the sampled calculators and found 12 defectives. • Determine a 95% confidence interval for the proportion of defectives in the entire population!



 p  p

P • Where: p

 p  z 

k 12   0.08  8 % n 150

p  (1-p)  1.96  n

0.08  (1-0.08)  1.96  0.022  0.043  4.3% 150

CI: 0.08 ± 0.043  [0.037; 0.123] = [3.7 % ; 12.3 % ]



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Statistical Inference

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