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- 1. INTRODUCTORY STATISTICS 1 Dilshod Achilov (Tajikistan)
- 2. TOPIC: Hypothesis testing 2
- 3. INFERENCE IN STATS Statistical Inference – it is a the process of drawing conclusions about a population based on a sample information 3 Dilshod Achilov
- 4. DISTRIBUTIONS As sample size increases, histogram class widths can be narrowed such that the histogram eventually becomes a smooth curve 4 Dilshod Achilov
- 5. DISTRIBUTION SHAPES 5 Dilshod Achilov
- 6. STEP 1 Specify the hypothesis to be tested and the alternative that will be decided upon if this is rejected The hypothesis to be tested is referred to as the Null Hypothesis (labelled H0) The alternative hypothesis is labelled H1 For the earlier example this gives: 6 mg500: mg500:0 aH H
- 7. STEP 1 (CONTINUED) The Null Hypothesis is assumed to be true unless the data clearly demonstrate otherwise 7 Dilshod Achilov
- 8. STEP 2 Specify a test statistic which will be used to measure departure from where is the value specified under the Null Hypothesis, e.g. in the earlier example. For hypothesis tests on sample means the test statistic is: 8 00 : H 0 5000 n s x t 0 Dilshod Achilov
- 9. STEP 2 The test statistic is a ‘signal to noise ratio’, i.e. it measures how far is from in terms of standard error units The t distribution with df = n-1 describes the distribution of the test statistics if the Null Hypothesis is true In the earlier example, the test statistic t has a t distribution with df = 25 9 n s x t 0 x 0 Dilshod Achilov
- 10. STEP 3 = 0.05 gives cut-off values on the sampling distribution of t called critical values values of the test statistic t lying beyond the critical values lead to rejection of the null hypothesis For the earlier example the critical value for a t distribution with df = 25 is 2.06 10
- 11. 11 t distribution with df=25 showing critical region 0 0.1 0.2 0.3 0.4 Density -4 -3 -2 -1 0 1 2 3 4 t Overlay Y's Y t distribution (df =25) Area t critical Overlay Plot critical values critical region 0.025 0.025
- 12. STEP 4 Calculate the test statistic and see if it lies in the critical region For the example t = -4.683 is < -2.06 so the hypothesis that the batch potency is 500 mg/tablet is rejected 12 683.4 26 783.10 500096.490 t
- 13. P VALUE The P value associated with a hypothesis test is the probability of getting sample values as extreme or more extreme than those actually observed, assuming null hypothesis to be true 13
- 14. 14 P value (contd.) 0 0.1 0.2 0.3 0.4 -5 -4 -3 -2 -1 0 1 2 3 4 5 t Overlay Y's Overlay Plot -4.683 4.683
- 15. TWO-TAIL AND ONE-TAIL TESTS The test described in the previous example is a two-tail test The null hypothesis is rejected if either an unusually large or unusually small value of the test statistic is obtained, i.e. the rejection region is divided between the two tails 15
- 16. ONE-TAIL TESTS Reject the null hypothesis only if the observed value of the test statistic is Too large Too small In both cases the critical region is entirely in one tail so the tests are one- tail tests 16

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