Chapter 1 Hypothesis Testing


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1 Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1
2 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population, for example the mean or the variance of a normal population. They may also concern the type, nature or probability distribution of the population. Statistical hypotheses are based on the concept of proof by contradiction. For example, say, we test the mean (m) of a population to see if an experiment has caused an increase or decrease in m. We do this by proof of contradiction by formulating a null hypothesis. 2
3 Elements of statistical hypothesis There are five ingredients to any statistical test : (a) Null Hypothesis (b) Alternate Hypothesis (c) Test Statistic (d) Rejection/Critical Region (e) Conclusion 3
4 Null Hypothesis It is a hypothesis which states that there is no difference between the procedures and is denoted by H0. For the above example the corresponding H0 would be that there has been no increase or decrease in the mean. Always the null hypothesis is tested, i.e., we want to either accept or reject the null hypothesis because we have information only for the null hypothesis. 4
5 Alternative Hypothesis It is a hypothesis which states that there is a difference between the procedures and is denoted by H1. 5
6 Various types of H0 and H1 Case Null Hypothesis H 0 Alternate Hypothesis H 1 1 µ 1 = µ 2 µ 1 µ 2 2 µ 1 < µ 2 µ 1 > µ 2 3 µ 1 > µ 2 µ 1 < µ 2 6
7 Reason for Rejecting H 0 Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean.... Therefore, we reject the null hypothesis that µ = µ = 50 H 0 Sample Mean 7
8 Test Statistic It is the random variable X whose value is tested to arrive at a decision. The Central Limit Theorem states that for large sample sizes (n > 30) drawn randomly from a population, the distribution of the means of those samples will approximate normality, even when the data in the parent population are not distributed normally. A z statistic is usually used for large sample sizes (n > 30) but often large samples are not easy to obtain, in which case the t distribution can be used. The population standard deviation s is estimated by the sample standard deviation, s. The t curves are bell shaped and distributed around t=0. t = X S n µ 8
9 Rejection Region It is the part of the sample space (critical region) where the null hypothesis H0 is rejected. The size of this region, is determined by the probability (α) of the sample point falling in the critical region when H0 is true. α is also known as the level of significance, the probability of the value of the random variable falling in the critical region. Also it should be noted that the term "Statistical significance" refers only to the rejection of a null hypothesis at some level α. It implies only that the observed difference between the sample statistic and the mean of the sampling distribution did not occur by chance alone. 9
10 Conclusion Conclusion : If the test statistic falls in the rejection/critical region, H0 is rejected, else H0 is accepted. 10
11 Errors Type I Error Reject True Null Hypothesis ( False Positive ) Has Serious Consequences Probability of Type I Error Is α Called Level of Significance Set by researcher Type II Error Do Not Reject False Null Hypothesis ( False Negative ) Probability of Type II Error Is β (Beta) 11
12 Errors Test Result True State H 0 True H 0 True Correct Decision H 0 False Type I Error H 0 False Type II Error Correct Decision α = P ( Type I Error) β = P( Type II Error) Goal: Keep α, β reasonably small 12
13 Level of Significance, α and the Rejection Region H 0 : µ 3 H 1 : µ < 3 α Critical Value H 0 : µ 3 H 1 : µ > 3 H 0 : µ = 3 H 1 : µ 3 Rejection Regions α α/2 13
14 Steps 1. State the null and alternative hypotheses 2. Choose α. The value should be small, usually less than 10%. It is important to consider the consequences of both types of errors. 14
15 3. Select the test statistic. Determine its value from the sample data. This value is called the observed value of the test statistic.! Remember that a t statistic is usually appropriate for a small number of samples; for larger number of samples, a z statistic can work well if data are normally distributed. 15
16 4 Compare the observed value of the statistic to the critical value obtained for the chosen α. 16
17 5. Make a decision: If statistic falls in the rejection region Reject H0 in favour of H1. If the test statistic does not fall in the critical region: Conclude that there is not enough evidence to reject H0. 17
18 Hypothesis Testing on mean Assumptions ttest: s Unknown Population is normally distributed If not normal, only slightly skewed & a large sample taken (Central limit theorem applies) Parametric test procedure t test statistic, with n1 degrees of freedom t = X S µ n 18
19 Hypothesis Testing: Steps Test the Assumption that the true mean of monthly cinema attendance of students is at least State H 0 H 0 : µ State H 1 H 1 : µ < Choose α α = Choose n n = Choose Test: t Test (or p Value) 19
20 Hypothesis Testing: Steps (continued) 6. Set Up Critical Value(s) t = Collect Data 25 students sampled, mean=2.7, s= Compute Test Statistic Computed Test Stat.= 2 (computed P value=.04, twotailed test) 9. Make Statistical Decision Reject Null Hypothesis 10. Express Decision The true mean is less than
21 Hypothesis Testing on σ 2 Twotailed test One tail test H o : σ 2 2 = σ o H a : σ 2 2 σ o Test statistic: χ 2 = (n  1)s 2 σ 2 o H o : σ 2 2 >= σ o H a : σ 2 2 > σ o reject H o if χ 2* 2 > χ α, n1 H o : σ 2 2 <= σ o H a : σ 2 2 < σ o reject H o if χ 2* 2 < χ α, n, n1 21
22 Homework & References 1. Voineagu, V. si colectiv Teorie si practica econometrica, Ed. Meteor Press, 2007, pages Read the text and solve the exercises. 2. David Ray Anderson,Dennis J. Sweeney,Thomas Arthur Williams,Thomas A. Williams  Statistics for business and economics, Chapter 9. pg=pa388&dq=hypothesis+testing+textbook+sweeney&sou rce=bl&ots=24jboaksnm&sig=xfpsccf1gcwalizya7bdjsp z80k&hl=ro&ei=1s29tk2omoohossgygy&sa=x&oi=book_r esult&ct=result&resnum=1&ved=0cbqq6aewaa#v=onepag e&q&f=false 22
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