Lecture_Hypothesis_Testing statistics .pptx

Hypothesis testing and descriptive statistics

1. Statistical Inference and Hypothesis Testing by Dr. Priyanka Dixit TISS, Mumbai

2. Descriptive and Inferential Statistics Descriptive statistics is the term given to the analysis of data that helps describe, show or summarize data in a meaningful way such that, pattern might emerge from the data. It do not, however, allow us to make conclusions beyond the data we have analysed or reach conclusions regarding any hypotheses we might have made. It is applicable to properly describe data through statistics and graphs.

3. Inferential Statistics Inferential statistics are techniques that allow us to use these samples to make generalization about the populations from which the samples were drawn.

4. Statistical Inference The process of generalization in prescribed manner from a sample to its universe is known as Statistical Inference. Universe/ Population µ σ SAMPLE Population Parameters µ: Population mean σ: Population standard deviation Sample Statistic x: Sample mean s: Sample standard deviation X s

5. Statistical Inference Inductive Inference: Extension from particular to the general is called inductive inference. Inductive inference involves element of uncertainty in the conclusions.

6. Deductive Inference Deductive inference can be described as a method of deriving information from the accepted facts, involves no uncertainty in the conclusions. The conclusions reached by deductive inference are conclusive.

7. Population and Sample The population is an abstract term that refers to the totality of all conceptually possible observations, measurements or outcomes of some specified kind. The number of conceptually possible observations is called the size of the population. The size varies according to the population being investigated.

8. Contd… For example, a study of monthly income may be conducted at a district, state and country level. So, in the first case, the population will consist of the income of one district, all residents of the state in the second case and in the third case income of all citizens of the country. A population may be finite when it consists of a given number of observations and infinite when it includes infinite number of observations.

9. Sample A sample is a set of observations selected from the population. The number of observations included in the sample is called the size of the sample. In finite population, a random sample is obtained by giving every individual in the population an equal chance of being chosen. In case of infinite population, a sample is random if each observation is independent of every other observation.

10. Parameter/Statistics Population and samples are studied through their characteristics. The most important of these characteristics are the Mean, the Variance and the Standard deviation. The characteristics of a population are called parameters. The characteristics of sample are called statistics.

11. Parameters (Population) Statistics (Sample) Population Mean Sample Mean Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation

12. The purpose of statistical inference is to make a judgment about the particular parameters on the basis of sample statistics. The judgment relating to population parameters are of two types; one is related to estimation of a parameter, the other with testing hypothesis about the parameter.

13. Hypothesis Testing Hypothesis testing in inferential statistics involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population. The hypothesis is tested against the information provided by sample in the form of a test-statistic. What is Statistical Hypothesis? A Hypothesis is a statement about one or more population parameters.

14. Null Hypothesis What is null hypothesis? A null hypothesis (H ) is a hypothesis of no relationship or no difference. Steps in hypothesis testing State the Hypothesis Set the criterion for rejecting H Compute the test statistic Decide whether to reject H

15. 1. State the Hypothesis In inferential statistics, the term hypothesis has a very specific meaning: conjecture about one or more population parameters. The hypothesis to be tested is called the null hypothesis and is given the symbol H . Example: We use a null hypothesis that the mean quantitative GRE score of the population of MPH students is 455. Thus, our null hypothesis, written in symbols, is H : µ = 455 OR H : µ-455 = 0 Where µ = population mean 455= Hypothesis value to be tested

16. We test the null hypothesis (H ) against the alternative hypothesis (symbolized H 1 ), which includes the possible outcomes not covered by the null hypothesis. For the above example we will use the alternative hypothesis as H 1 : µ ≠ 455 The alternative hypothesis, often considered the research hypothesis, can be supported only be rejecting the null hypothesis.

17. 2. Set the Criterion for Rejecting H After stating the hypothesis the next step in hypothesis testing is determining how different the sample statistic must be from the hypothesized population parameter (µ) before the null hypothesis can be rejected. For our example, suppose we randomly select 144 MPH students from the population and find the sample mean to be 535. Is this sample mean =535 sufficiently different from what we hypothesize for the population mean (µ = 455) to warrant rejecting null hypothesis. Before answering this question, we need to consider three concepts: (i) errors in hypothesis testing, (ii) level of significance, and (iii) Region of rejection

18. Properties of Normal Distribution 8. The areas of a normal curve are measured in standard deviation units. The proportions of cases in specified areas of a normal curve, as marked by standard deviations, are constant as detailed below: Number of standard Results lying outside deviation from mean this (%) 1.00 31.74 1.64 10.00 1.96 5.00 2.58 1.00 3.29 0.10

19. i. Errors in hypothesis testing When we decide to reject or not reject the null hypothesis, there are four possible situations: A true hypothesis is rejected. A true hypothesis is not rejected. A false hypothesis is not rejected A false hypothesis is rejected

20. In a specific situation, we may make one of two types of errors, as shown in the figure below: Decision made State of nature Null hypothesis is true Null hypothesis is false Reject null hypothesis Type I error Correct decision Do not reject null hypothesis Correct decision Type II error

21. Example Verdict of Jury Defendant Guilty Innocent Not Guilty Incorrect Correct decision Guilty Correct decision Incorrect

22. Contd… Errors Type I error is when we reject a true null hypothesis. Type II error is when we do not reject a false null hypothesis

23. ii. Level of significance To choose the criterion for rejecting H , the researcher must first select what is called the level of significance. The level of significance or alpha (α) level is defined as the probability of making a Type I error when testing a null hypothesis. The level of significance is the probability of making a Type I error: rejecting H when it is true.

24. Power of the Test Type II error involves acceptance of H when it is actually false or not finding an effect when actually there is an effect. β is the probability of type II error. (1-β) is called the power of the test= Probability of finding an effect when actually there is an effect. Power of a statistical test is analogous to the sensitivity of a diagnostic test. α being the false positive. β being the false negative.

25. iii. Region of Rejection The region of rejection is the area of the sampling distribution that represents those values of the sample mean that are improbable if the null hypothesis is true. The Critical values of the tests statistic are those values in the sampling distribution that represent the beginning of the region of rejection. When the alternative hypothesis is non-directional, the region of rejection is located in both tails of the sampling distribution. The test of the null hypothesis against this non-directional alternative is called a two-tailed test. The probability of obtaining a mean as extreme as or more extreme than the observed sample mean (xbar), given that the null hypothesis is true, is called the p-value of the test or p.

26. Properties of Normal Distribution 8 . Properties of Normal Distribution The areas of a normal curve are measured in standard deviation units. The proportions of cases in specified areas of a normal curve, as marked by standard deviations, are constant as detailed below: Number of standard Results lying outside deviation from mean this (%) 1.00 31.74 1.64 10.00 1.96 5.00 2.58 1.00 3.29 0.10

27. Region of rejection for sampling distribution of the mean for null hypothesis H : µ = 455 and S.D. (σ x ) = 8.33

28. 3. Compute the Test Statistic In our example µ=455, the hypothesized value for the parameter n=144, the size of the sample = 535, the observed value for the sample statistic σ=100, the value of the standard deviation in the population First using the concept of z scores, we determine how Different is from µ, or the number of standard errors (standard deviation units) the observed sample value is from the hypothesized value. In symbols,

29. calculating the z score using above formula is called computing the test statistic

30. 4. Decide about H Suppose we had found that the sample mean for 144 students was not 535, but 465. Our hypotheses, sampling distribution, and critical values (+1.96 and -1.96) remain the same, but now the test statistic is

31. In other words, the observed sample mean ( = 465) is 1.20 standard errors above the hypothesized value of the population mean.

32. Theoretical sampling distribution for the hypothesis H :µ=45, illustrating the values of the test statistic when =465 Note that the test statistic (1.20) does not exceed the critical value; it does not fall into the region of the rejection; and we should not reject the null hypothesis . -1.96 +1.96 1.20 9.60

33. This test statistic (1.20) is then compared to the critical value (1.96). If the test statistic exceeds the critical values in absolute value, then the null hypothesis is rejected. If the test statistic does not exceeds the critical values in absolute value, then the null hypothesis is accepted.

34. Region of rejection : Directional Alternative Hypothesis In the GRE example, we tested the null hypothesis against a non-directional alternative: H : µ = 455 H 1 : µ ≠ 455 This test is called two-tailed or non-directional because the region of rejection was located in both tails of the sampling distribution of the mean. Suppose a direction of the results is anticipated. A directional hypothesis states that a parameter is either greater or less than the hypothesis value. For instance, in the GRE example we might use the alternative hypothesis that the mean GRE level of our population is greater than 455, in symbols, H : µ = 455 H 1 : µ > 455

35. An alternative hypothesis can be either non-directional or directional. A directional alternative hypothesis states that the parameter is greater than or less than the hypothesized value. A non-directional alternative hypothesis merely states that the parameter is different from (not equal to) the hypothesized value.

36. The test of the null hypothesis against a directional alternative is called a one-tailed test, the region of rejection is located in one of the two tails of the sampling distribution. The specific tail of the distribution is determined by the direction of the alternative hypothesis. Now suppose the alternative hypothesis states that the mean GRE was less than 455. In symbols, the hypotheses are H : µ = 455 H 1 : µ < 455 Here the critical region lies on the left tail of the distribution.

37. Type-I and Type-II Errors in Decision Making In a specific situation, we may make one of two types of errors, as shown in the figure below: Decision taken by the investigator Existing Reality Group A=Group B Group A # Group B Group A # Group B P[ Type-I Error] (Level of significance) Correct Decision (Power of the study) Group A=Group B Correct Decision (Level of confidence) Type – II Error

38. Testing of Hypothesis Q=1 A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 100. Test the null hypothesis that µ=100 against the alternative hypothesis (µ≠100) using α=0.05. (b) Test the null hypothesis that µ=100 against the alternative hypothesis (µ>100) using α=0.05

39. Testing of Hypothesis Ex=1 A random sample of 200 observations from a population with standard deviation 80 yielded a sample mean of 150. Test the null hypothesis that µ=100 against the alternative hypothesis (µ≠100) using α=0.05. (b) Test the null hypothesis that µ=100 against the alternative hypothesis (µ>100) using α=0.05

40. Ex=2 A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 100. (a)Test the null hypothesis that µ=111 against the alternative hypothesis (µ≠111) using α=0.05. (b) Test the null hypothesis that µ<=111 against the alternative hypothesis (µ>111) using α=0.05 Explain why the results differ.

41. Q=2 The heights of 10 males of a given locality are found to be as follows: 70, 67, 62, 68, 61, 68, 70, 64, 64, 66 inches. Is it reasonable to believe that the average height is greater than 64 inches? What will be the finding if alternative hypothesis was two-tailed

42. Contd.. Answer Mean=66; S.D.=3.16 and Variance=10.00, t=2.00 The tabulated value of t-statistic at 9 d.f. and α=0.05 (one-tailed) is 1.833 Since calculated value is greater than the tabulated value, we will reject the null hypothesis . We can believe that mean height is greater than 64 inches. What will be the finding if alternative hypothesis was two-tailed (answer it).

43. Student’s t Distributions Does the adjustment of using s to estimate σ have an effect on the statistical test? Actually, it does, especially for small samples. The effect is that the normal distribution is inappropriate as the sampling distribution of the mean. In the beginning of the 20 th century William S. Gosset found that, for small samples, sampling distribution departed substantially from the normal distribution and that, as sample sizes changed, the distributions changed. This gave rise to not one distribution but a family of distributions. The t distributions are a family of symmetrical, bell-shaped distributions that change as the sample size changes.

44. Degrees of Freedom Degrees of Freedom : The number of degrees of freedom is a mathematical concept defined as the number of observations less the number of restrictions placed on them.

45. Student’s t distribution for 1, 2, 5, 10, and ∞ degrees of freedom

46. Point Estimates and Interval Estimates A point estimate is a single value that represent the best estimate of the population value. If we are estimating the mean of a population (µ), then the sample mean is the best point estimates. Interval Estimation builds on points estimation to arrive at a range of values that are tenable for the parameter and that define an interval we are confident contains the parameter.

47. Confidence Interval CI= ± (Z CV ) (σ X ) Where = Sample mean Z CV = Critical value using the normal distribution and σ X = Standard error of the mean

48. Confidence Interval CI= ± (t CV ) (s X ) Where = Sample mean t CV = Critical value using appropriate t distribution and s X = estimated standard error of the mean from the sample

49. Comparison of Two Means Q=As part of an investigation of the development of infant sleep patterns, the sleep of 20 infants (10 male and 10 female) was monitored on several occasions between 1 week and 6 months of age. The quiet sleep results (in minutes) at 1 week of age for the 20 study infants follow. Is there evidence of a difference in quiet sleep behavior between two genders? Is there evidence that male mean quiet sleep behavior is higher than female? Quiet sleep (male) 85 129 215 143 44 173 230 198 105 127 Mean=144.90 Quiet sleep (female) 140 155 33 209 166 72 116 131 97 124 Mean=124.30

50. Sp is pooled variance, Sm^2 and Sf^2 is variance of two sample set

51. Contd… Answer For male; S 1 =59.35; S 1 2 =3522.54; Mean=144.90 For female; S 2 =49.48; S 2 2 =2448.011; Mean=124.30 t=0.843 at 18 d.f.

52. Paired-t-test As part of a study to determine the effects of a certain oral contraceptive on weight gain; nine healthy females were weighted at the beginning of a course of oral contraceptive use. They were reweighed after 3 months. Results are given below. Do the results suggest evidence of weight gain? Longitudinal Study/Real-Cohort Study Subject Initial weight (LBS) 3 - Months weight (LBS) 1 120 123 2 141 143 3 130 140 4 150 145 5 135 140 6 140 143 7 120 118 8 140 141 9 130 132

53. Contd… Answer t=1.509 One-tailed Tabulated value of t at α=0.05 and d.f. =8 is 1.860 (one-tailed).

54. Male Female 42.1 41.3 42.4 43.2 41.8 42.7 43.8 42.5 43.1 44.0 41.0 41.8 42.8 42.3 42.7 43.6 43.3 43.5 41.7 44.1 Do the data provide sufficient evidence to conclude that, on the average, the male weight is greater than female weight? Perform the required hypothesis test at the 5% level of significance.

55. Proportion Test Q=1 In a sample of 1000 people in Maharashtra, 540 are rice eaters and the rest are wheat eaters. Can we assume that both rice and wheat are equally popular in this state at 1% level of significance? Z tabulated at 1% level of significance is 2.58 (two-tailed). Q=2 Twenty people were attacked by a disease and only 18 survived. Will you reject the hypothesis that the survival rate, if attacked by this disease, is 85% in favour of the hypothesis that it is more, at 5% level. Z tabulated at 5% level of significance is 2.58 (one-tailed).

56. Q=3 In a year there are 956 births in a town A of which 52.5% were males, while in towns A and B combined, this proportion in a total of 1406 births was 0.496. Is there any significant difference in the proportion of male births in the two towns? Z tabulated at 5% level of significance is 1.96 (two-tailed).

57. References Medical Statistics-Principles & Methods by K.R. Sundaram, S. N. Dwivedi and V Sreenivas.