Estimation and hypothesis

Infretial statistics which consist of Estimation and hypothesis

1. Estimation and Hypothesis Testing for sample Mean

2. Group Members Junaid Ijaz Wajahat Saadat Asfand Yar Tahir Sohaib Arshad Daud Amir Salman A bbasi Aqib Sharif

3. Content Inferential statistics Estimation Application of Estimation Hypothesis Application of Hypothesis References

4. STATISTICAL ANALYSIS DESCRIPTIVE INFERENTIAL L NUMERICAL GRAPHICAL Estimation Hypothesis testing Point estimate Interval estimate

5. Inferential statistics The part of statistics that allows researchers to generalize their findings to a larger population by using data from the sample collected. Estimation of parameters * Point Estimation * Intervals Estimation – Hypothesis Testing Two ways to make inference:

6. Parameter For Sample For Population Mean: X  Standard deviation: s  Proportion : p 

7. Estimation The process by which one makes inferences about a population, based on information obtained from a sample . Point estimate Interval estimate

8. Point estimate Point estimates are single points that estimates parameter directly which serve as a " best guess " or " best estimate " of an unknown population parameter • sample proportion pˆ (“p hat”) is the point estimate of p • sample mean x (“x bar”) is the point estimate of μ • sample standard deviation s is the point estimate of σ

9. Criteria of a good estimator: There can be more than one estimator for the parameter. A good estimator is unbiased i.e E(T )=  Sample mean is an unbiased estimator of population mean . The difference between two sample means is an unbiased estimate of difference between the population mean.

10. Disadvantages of point estimates Point estimate do not provide ( i ) Information about sample to sample variability (ii) How precise is x(Sample mean) as an estimate of μ(Population mean) (iii) How much can we expect x vary from μ

11. Interval Estimation: In interval estimation, an interval is constructed around the point estimate, and it is stated that this interval is likely to contain the corresponding population parameter. In general an interval estimate may be expressed as follows margin of error estimator  (reliability coefficient)  (standard error)  z ( 1   / 2 )  x  

12. Standard Error The standard error(SE) is very similar to  standard deviation . Both are measures of spread. The higher the number, the more spread out your data is. To put it simply, the two terms are essentially equal — but there is one important difference. While the standard error uses  statistics  (sample data) standard deviations use  parameters  (population data). is the probability that the interval computed from the sample data includes the population parameter of interest Confidence level : n  ( 1- a ) 100%

13. Process for Constructing Confidence Intervals Compute the sample statistic (e.g. a mean) Compute the standard error of the mean Make a decision about level of confidence that is desired (usually 95% or 99%) Find tabled value for 95% or 99% confidence interval Multiply standard error of the mean by the tabled value Form interval by adding and subtracting calculated value to and from the mean

14. Z- Distribution When Population parimeters (Standard daviation ) Are Known Then we use confidence level to find value of reliability coefficient From table of standard normal distribution T - Distribution When Population parimeters (Standard daviation ) Are not Known Then we use confidence level and degree of fredom to find value of reliability coefficient From table of T-distribution

15. FACTORS AFFECTING CONFIDENCE INTERVAL (a)Level of confidence: Greater the level of confidence greater will be the interval (b)Sample size (N): Greater the sample size greater will be the interval (c)Data variability(  ):  value of a sa mple is not under the control of the investigator. Hence, the width of a confidence interval cannot be controlled using 

16. Application of Estimation in Civil Engineering

17. COST ESTIMATION

18. ESTIMATING MATERIALS ESTIMATING Project TIME ESTIMATING LABOR

19. Environmental Engineering

20. ESTIMATING ANNUAL INCREASE IN TEMPERATURE GLOBAL WARMING ANNUAL RAINFALL ESTIMATING PARAMETER CONTRIBUTING TO POLUUTION

21. URBAN PLANNING

22. A city planning estimate is the estimation of the maximum permissible construction parameters, conditions of combining the architectural planning and maintenance system, engineering communications, and transport service. It may include estimation of population parameter, water requirement and drainage.

23. TRAFFIC ENGINEERING

24. ESTIMATING TRAFFIC DENSITY: Traffic density is a major congestion indicator, and because its measurement is difficult, it is usually estimated from other readily measurable parameters. Traffic flow model is formulate, and using this model, a model-based estimation scheme is designed.

25. What is a Hypothesis? I assume the mean SBP of An assumption about population is 120 mmHg the population parameter .

26. Hypothesis Hypothesis is a predictive statement, capable of being tested by scientific methods, that relates an independent variables to some dependent variable. A hypothesis states what we are looking for and it is a proportion which can be put to a test to determine its validity e.g. Students who receive counseling will show a greater increase in creativity than students not receiving counseling

27. Characteristics of Hypothesis Clear and precise. Capable of being tested. Stated relationship between variables. limited in scope and must be specific. Stated as far as possible in most simple terms so that the same is easily understand by all concerned. But one must remember that simplicity of hypothesis has nothing to do with its significance. Consistent with most known facts. Responsive to testing with in a reasonable time. One can’t spend a life time collecting data to test it. Explain what it claims to explain; it should have empirical reference.

28. Types of Hypothesis Null Hypothesis Alternative Hypothesis

29. Null Hypothesis It is an assertion that we hold as true unless we have sufficient statistical evidence to conclude otherwise. Null Hypothesis is denoted by 𝐻 If a population mean is equal to hypothesised mean then Null Hypothesis can be written as 𝐻 : 𝜇 = 𝜇

30. Alternative Hypothesis The Alternative hypothesis is negation of null hypothesis and is denoted by 𝐻 𝑎 If Null is given as 𝐻 : 𝜇 = 𝜇 Then alternative Hypothesis can be written as 𝐻 𝑎 : 𝜇 ≠ 𝜇 𝐻 𝑎 : 𝜇 > 𝜇 𝐻 𝑎 : 𝜇 < 𝜇

31. Level of significance and confiden c e Significance means the percentage risk to reject a null hypothesis when it is true and it is denoted by 𝛼 . Generally taken as 1%, 5%, 10% ( 1 − 𝛼) is the confidence interval in which the null hypothesis will exist when it is true.

32. Two tailed test @ 5 % Significance level Acceptance and Rejection regions in case of a Two tailed test 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 / 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.025 𝑜𝑟 2.5%) 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.025 𝑜𝑟 2.5%) Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 ≠ 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝐻 : 𝜇 = 𝜇

33. Left tailed test @ 5% Significance level Acceptance and Rejection regions in case of a left tailed test 𝐻 : 𝜇 = 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.05 𝑜𝑟 5%) Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 < 𝜇

34. Right tailed test @ 5 % Significance level Acceptance and Rejection regions in case of a Right tailed test Suitable When 𝐻 : 𝜇 = 𝜇 𝐻 𝑎 : 𝜇 > 𝜇 𝑇𝑜𝑡𝑎𝑙 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑜𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (1 − 𝛼) = 95% 𝐻 : 𝜇 = 𝜇 𝑅𝑒𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑔𝑖𝑜𝑛 /𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 (𝛼 = 0.05 𝑜𝑟 5%)

35. Proce dur e for Hyp o thesis Te s t i n g State the null (Ho)and alternate (Ha) Hypothesis State a significance level; 1%, 5%, 10% etc. Decide a test statistics; z-test, t- test . Calculate the value of test statistics Do the required calculations Define the critical region through z table If z fall in critical region If z doesn’t fall in the critical region Accept Ho Reject Ho

36. Hy p othesis Testing of Means Z-TEST T-TEST

37. Z-Test for testing means Test Condition  Population normal and infinite  Sample size large or small,  Population variance is known  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑧 = 𝜎 𝑝 𝑛

38. Z-Test for testing means Test Condition  Population is infinite and may not be normal,  Sample size is large,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑧 = 𝜎 𝑠 𝑛

39. T-Test for testing means Test Condition  Population is infinite and normal,  Sample size is small,  Population variance is unknown  Ha may be one-sided or two sided Test Statistics 𝑋 −𝜇 𝐻 𝑡 = 𝜎 𝑠 𝑛 𝑤𝑖𝑡ℎ 𝑑. 𝑓. = 𝑛 − 1 𝜎 𝑠 = 𝑋 𝑖 − 𝑋 2 (𝑛 − 1)

40. Type I and Type I I Error Situation Decision Accept Null Reject Null Null is true Correct Type I error ( 𝛼 𝑒𝑟𝑟𝑜𝑟 ) Null is false Type II error ( 𝛽 𝑒𝑟𝑟𝑜𝑟 ) Correct

41. Limitations of the test of Hypothesis Testing of hypothesis is not decision making itself; but help for decision making Test does not explain the reasons as why the difference exist, it only indicate that the difference is due to fluctuations of sampling or because of other reasons but the tests do not tell about the reason causing the difference. Tests are based on the probabilities and as such cannot be expressed with full certainty. S tatistical inferences based on the significance tests cannot be said to be entirely correct evidences concerning the truth of the hypothesis.

42. Practical Uses of Hypothesis Testing

43. In Civil Engineering A failure of civil structure can cause fatal destructions which could have devastating effects on contractor’s firm and engineer’s career. Engineers can predict the outcomes of any hypothesis precisely by using statistical hypothesis testing. Data like seismic activity, annual precipitation, rate of silt deposition in dams etc. can be used in hypothesis testing.

44. Human Gender Ratio  It is the ratio of males to the females in a population. Gender Imbalance may leads to many problems that are prevailing in some parts of the world. Consequences of Gender imbalance may include: Rapid decline in fertility Differential marital statistics Increase in antisocial behavior and violence

45. The earliest use of statistical hypothesis testing is to the question of whether male and female births are equally likely (null hypothesis), which was addressed in the 1700s by John Arbuthnot (1710) John Arbuthnot, in 1710, was the first person who used hypothesis testing to question whether male and female births are equally likely or not. Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. He concluded that every year, the number of males born in London exceeded the number of females. The probability of the observed outcome is 0.5 82 .

46. Laplace, 1770, considered the statistics of almost half a million births. The statistics showed an excess of boys compared to girls.  He concluded by calculation of a  p -value that the excess was a real, but unexplained, effect.

47. Courtroom Trial A defendant is considered not guilty as long as his/her guilt is not proven. Only when there is enough evidence for the prosecution then the defendant is convicted. A criminal trial can be regarded as either or both of two decision processes: guilty vs not guilty or evidence vs a threshold. A hypothesis test can be regarded as either a judgment of a hypothesis or as a judgment of evidence.

48. Business Applications Generally firms proposes a monthly income investment plan that promises a variable return each month. The real challenge is taking precise decisions at appropriate times. Hypothesis Testing can make a businessman’s life easier.

49. Medical Applications Sometimes medics have to take risks while conducting operations or proposing a medicine which may have side reactions on a patient. A wrong decision can consume a life. By visualizing genetics data and reactions of medicine/surgery on various patients, doctors and surgeons can precisely take correct decisions.

50. References http:// stattrek.com/estimation/estimation-in-statistics.aspx http://www.cabrillo.edu/~ evenable/Ch08.pdf https://courses.lumenlearning.com/wmopen-concepts-statistics/chapter/estimating-a-population-mean-1-of-3 / https:// www.cliffsnotes.com/study-guides/statistics/principles-of-testing/point-estimates-and-confidence-intervals https:// www.cliffsnotes.com/study-guides/statistics/principles-of-testing/point-estimates-and-confidence-intervals

51. Thanks