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Azza Osman Mohamed

University of Khartoum Faculty of Mathematical Science Department of Information Technology. Applied Statistics احصاء تطبيقي (احص 301). Azza Osman Mohamed. Statistical Estimates. Test of Hypotheses . Correlation. Simple Linear Regression Analysis. Analysis of Variance.

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Azza Osman Mohamed

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  1. University of KhartoumFaculty of Mathematical Science Department of Information Technology Applied Statistics احصاء تطبيقي (احص 301) Azza Osman Mohamed

  2. Statistical Estimates. Test of Hypotheses . Correlation. Simple Linear Regression Analysis. Analysis of Variance. Non-parametric Test. Statistic package SPSS. التقدير الاحصائي. اختبارات الفروض. الارتباط الخطي. الانحدار الخطي البسيط. تحليل التباين. الاختبارات اللامعلمية. الحزمة الاحصائية SPSS Course componentمحتوى المقرر

  3. Course aim: The aim of this course is to develop further understanding of statistical methods. • Outcome: By the end of this course you will be able to: • Understandthe inferential statistics. • Describing common measures of correlation and association, and performing simple regression analysis. • understand the workings of the analysis of variance table and its application to one-way ANOVA, and two-way ANOVA situations. • understand the workings of the non-parametric methods. • Perform statistical analysis using SPSS. • Present and interpret the results. • Course evaluation: • Assignments. • Labs . • Mid-term exam. • Final exam.

  4. Session 1

  5. Learning Objectives • At the end of session 1 and 2 you will be able to • State Estimation Process • Introduce Properties of Point Estimates • Explain Confidence Interval Estimates • Compute Confidence Interval Estimation for Population Mean ( known and unknown) • Compute Confidence Interval Estimation for Population Proportion

  6. Introduction to Estimation Point Estimation

  7. Statistical Methods Statistical Methods Descriptive Inferential Statistics Statistics Hypothesis Estimation Testing

  8. Statistical Inference… Statistical inferenceis the process by which we acquire information and draw conclusions about populations from samples. • In order to do inference, we require the skills and knowledge of descriptive statistics, probability distributions, and sampling distributions.

  9. Inference Process Estimates & Tests Population Sample Statistics Sample X, Ps

  10. Thinking Challenge • Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

  11. Estimation Methods Estimation Point Interval Estimation Estimation

  12. Estimation… • The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic. • An estimator is a method for producing a best guess about a population value. • An estimate is a specific value provided by an estimator. • Example: We said that the sample mean is a good estimate of the population mean • The sample mean is an estimator • A particular value of the sample mean is an estimate

  13. Point Estimator… Definition: • A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point. • Gives no information about how close value is to the unknown population parameter • Example: the sample mean ( ) is employed to estimate the population mean ( ).

  14. Population Parameters Are Estimated with Point Estimator Estimate Population with Sample Parameter Statistic  Mean X Proportion p p s 2 2 Variance s  Differences   X X 1 2 1 2

  15. Point Estimator… • Question: Is there a unique estimator for a population parameter? For example, is there only one estimator for the population mean? • The answer is that there may be many possible estimators • Those estimators must be ranked in terms of some desirable properties that they should exhibit

  16. Properties of Point Estimators • The choice of point estimator is based on the following criteria • Unbiasedness • Efficiency • Consistency Unbiased Estimatorsعدم التحيز : Definition • A point estimator is said to be an unbiased estimator of the population parameter  if its expected value (the mean of its sampling distribution) is equal to the population parameter it is trying to estimate • We can also define the bias of an estimator as follows

  17. Properties of Point Estimators • To select the “best unbiased” estimator, we use the criterion of efficiency Efficiency:الكفاءة Definition • An unbiased estimator is efficient if no other unbiased estimator of the particular population parameter has a lower sampling distribution variance. • If and are two unbiased estimators of the population parameter , then is more efficient than if • The unbiased estimator of a population parameter with the lowest variance out of all unbiased estimators is called the most efficient or minimum variance unbiased estimator (MVUE).

  18. Properties of Point Estimators Consistency :الاتساق Definition: • We say that an estimator is consistent if the probability of obtaining estimates close to the population parameter increases as the sample size increases • One measure of the expected closeness of an estimator to the population parameter is its mean squared error • The problem of selecting the most appropriate estimator for a population parameter is quite complicated

  19. References….. • Inferences Based on a Single Sample: Estimation with Confidence Intervals John J. McGill/Lyn Noble Revisions by Peter Jurkat • Chapter 10 Introduction on to Estimation Brocks/Cole , a division of Thomson learning, Inc. • Basic Business Statistics: Concepts & ApplicationsChapter 8 -Confidence IntervalEstimation • Chapter 1, Point Estimation Algorithms , Department of Computer science, University of Tennessee ,USA

  20. Session 2

  21. Introduction to Estimation Interval Estimation

  22. Estimation Methods Estimation Point Interval Estimation Estimation

  23. Confidence Interval Estimation Process I am 95% confident that  is between 40 & 60. Population Random Sample Mean X = 50 Mean, , is unknown

  24. Interval Estimator… • An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. • Provide us with a range of values that we belive, with a given level of confidence, containes a true value. • That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds. • Gives Information about Closeness to Unknown Population Parameter Sample Statistic (Point Estimate) ConfidenceInterval Confidence Limit (Lower) Confidence Limit (Upper)

  25. Point & Interval Estimation… • For example, suppose we want to estimate the mean summer income of a class of IT students. For n=25 students, is calculated to be 400 $/week. point estimate interval estimate An alternative statement is: The mean income is between 380 and 420 $/week.

  26. Probability that the unknown population parameter θ falls within interval الفترة تسمي فترة الثقة للمعلمة θ. probability that “true” parameter  is in the interval is equaled to 1-. 1-  is called confidence level. 1-  يسمى معامل الثقة وهو احتمال احتواء الفترة على المعلمة θ . Limits of the interval are called lower and upper confidence limits. Confidence Interval (CI)..فترة الثقة ...

  27. Confidence Interval (CI)..فترة الثقة ... • Actual realization of this interval is called a (1- )% 100 of confidence interval. • نكون واثقين بمقدار (1- )% 100 بأن المعلمة المجهولة تقع داخل الفترة . • We are 95% confident that the 95% confidence interval will include the population parameter • a (5%)is probability that parameter is Notwithin interval • Typical values are 99%, 95%, 90%, …

  28. Interval and Level of Confidence Sampling Distribution of the Mean Intervals extend from to of intervals constructed contain  ;  100% do not. Confidence Intervals

  29. Know Central Intervals of the Normal Distribution  +1.65x +2.58x -1.65x -2.58x -1.96x +1.96x 90% Confidence 95% Confidence 99% Confidence X=  ± Zx 

  30. Factors Affecting Interval Width • 1. Data Dispersion • Measured by sX • 2. Sample Size • s`X = sX / Ön • 3. Level of Confidence (1 - a) • Affects Z Intervals Extend from`X - Zs`X to`X + Zs`X

  31. Confidence Interval Estimates Confidence Intervals Mean Proportion Variance s Known s Unknown x x

  32. Estimating μwhen σ is known… Known, i.e. sample mean Known, i.e. standard normal distribution Unknown, i.e. we want to estimate the population mean Known, i.e. its assumed we know the population standard deviation… Known, i.e. the number of items sampled

  33. Confidence Interval Estimator for μ Confidence Interval Estimator for μ Usually represented with a “plus/minus” ( ± ) sign upper confidence limit (UCL) lower confidence limit (LCL)

  34. Four commonly used confidence levels… • Confidence Level

  35. Example … • A computer company samples demand during lead time over 25 time periods: • Its is known that the standard deviation of demand over lead time is 75 computers. We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…

  36. Example … • “We want to estimate the mean demand over lead time with 95% confidence in order to set inventory levels…” • Thus, the parameter to be estimated is the pop’n mean μ. • And so our confidence interval estimator will be:

  37. Example … • In order to use our confidence interval estimator, we need the following pieces of data: • therefore: • The lower and upper confidence limits are 340.76 and 399.56. Calculated from the data… Given

  38. Thinking Challenge • The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for mX ifs2X = 100.

  39. What is interval for sample size = 100?

  40. Confidence Interval Estimates Confidence Intervals Mean Proportion Variance s Known s Unknown x x

  41. Confidence Interval for Mean of a Normal Distribution with Unknown Variance • If the sample size is large n 30 ≤ : في حالة حجم العينة كبير • The population variance is not be known • The sample standard deviation will be a sufficiently good estimator of the population standard deviation • Thus, the confidence interval for the population mean is:

  42. Confidence Interval for Mean of a Normal Distribution with Unknown Variance • If the sample size is small and the population variance is unknown, we cannot use the standard normal distribution • If we replace the unknown  with the sample st. deviation s the following quantity follows Student’s t distribution with (n – 1) degrees of freedom • The t-distribution has mean 0 and (n – 1) degrees of freedom • As degrees of freedom increase, the t-distribution approaches the standard normal distribution

  43. Student’s t Distribution Estimates the distribution of the sample mean, , when the distribution to be sample is normal Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) Z t 0

  44. Confidence Interval for Mean of a Normal Distribution with Unknown Variance • a 100(1-)% confidence interval for the population mean  when we draw small samples from a normal distribution with an unknown variance 2 is given by

  45. Student’s t Table  / 2 v t t t .10 .05 .025 12.706 1 3.078 6.314 2 1.886 2.920 4.303 3 1.638 2.353 3.182  / 2 0 t t t values Assume:n = 3df = n - 1 = 2 = .10/2 =.05 2.920

  46. Estimation Example Mean ( Unknown) A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for . with 95% confidence

  47. Thinking Challenge For a sample where the sample size = 9, the sample mean = 28 and the sample s.d. = 3. What is the closest 95% confidence interval of the mean? Select A for [27, 29] B for [26.5, 29.5] C for [26, 30] D for [25.25, 30.75] E for [24.5, 31.5]

  48. If we want to estimate the population proportion and n is large then: اذا كان من المتوقع ان لا تكون نسبة النجاح غير معلومة وكان حجم العينة كبير فإن : and Where x is the number of success . Confidence Interval For the Population Proportion • Confidence interval estimate

  49. A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p. Example …. with 95% confidence

  50. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

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