1036: Probability & Statistics

Transcription

1 036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0-

2 Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso s blood type or eye color are idepedet variables. A statistical hypothesis is a assertio or cojecture cocerig oe or more populatios. True of False is ever kow with absolute certaity uless the etire populatio is examied. The decisio procedure is doe with the awareess of the probability of a wrog coclusio 0-2

3 Role of Probability i Hypothesis Testig The acceptace of a hypothesis merely implies that the data do ot give sufficiet evidece to refute it. Rejectio meas that there is a small probability of obtaiig the sample iformatio observed whe the hypothesis is true. Example: for the cojecture of the fractio defective p 0.0, a sample of 00 revealig 20 defective items is certaily evidece of rejectio Sice the probability of obtaiig 20 defectives is approximately The firm coclusio is established by the data aalyst whe a hypothesis is rejected 0-3

4 Supportig a Cotetio To reject the hypothesis Cotetio: coffee drikig icreases the risk of cacer Hypothesis: there is o icrease i cacer risk produced by drikig coffee Cotetio: oe kid of gauge is more accurate tha aother Hypothesis: there is o differece i the accuracy of the two kids of gauges 0-4

5 Null ad Alterative Hypotheses Structure of hypothesis Null hypothesis, H 0 ay hypothesis we wish to test Alterative hypothesis, H the opposite hypothesis to reject H 0 Example H 0 is the ull hypothesis p 0.5 for a biomial populatio, H would be oe of the followig: p > 0.5, p < 0.5, or p

6 Testig a Statistical Hypothesis Rejectio of the ull hypothesis whe it is true is called a type I error level of sigificace. Acceptace of the ull hypothesis whe it is false is called a type II error. The probability of committig a type I error is deoted by α The probability of committig a type II error is deoted by β 0-6

7 0-7 Example: α ad β ;20, whe type II error ;20, 8 whe type II error ;20, ;20, 8 whe error type I > x x x x x b p P P x b p P P x b x b p P P β β α H 0 : x 8; H : x > 8 p /4, 20, biomial

8 Remarks Critical value: the last umber passig from the acceptace regio ito the critical regio For a fixed sample size: A reductio i β is always possible by icreasig the size of the critical regio A decrease i the probability of oe error usually results i a icrease i the probability of the other error The probability of committig both types of error ca be reduced by icreasig the sample size 0-8

9 The Role of α, β, ad Sample Size To determie the probability of committig a type I error, we shall use the ormal-curve approximatio with > 30. Example: H 0 : p /4; H : p > /4 00, critical value 36 p 00 α PtypeI error P P Z > p , 50, β PtypeIIerror P P Z < pq 00 > 36whe p pq 00 36whe p , z P Z > , z

10 Hypothesis Testig with a Cotiuous Radom Variable Cosider the ull hypothesis that the average weight of male studets i a certai college is 68 kilograms agaist the alterative hypothesis that it is uequal to 68. H 0 : 68; H : , 36, α Ptype I error / P z.67, z α P Z <.67 + P Z >.67 2P Z < icrease sample size to 64, 3.6 /6 0.6 < 67 whe 68 + P 3.6 / z 2.22, z α P Z < P Z > P Z < > 69 whe

11 Hypothesis Testig with a Cotiuous Radom Variable 3.6, 36, β PtypeII error P67 < 69 whe z 6.67, z β P 6.67 < Z < 2.22 P Z < P Z / 3.6/6 0.6 < If thealterative hypothesis z 3.33, z β P 3.33< Z <. P Z <. P Z <

12 Properties of a Test Hypothesis The type I error ad type II error are related. A decrease i the probability of oe geerally results i a icrease i the probability of the other The size of the critical regio, ad therefore the probability of committig a type I error, ca always be reduced by adjustig the critical values. A icrease i the sample size will reduce ad simultaeously If the ull hypothesis is false, is a maximum whe the true value of a parameter approaches the hypothesized value. The greater the distace betwee the true value ad the hypothesized value, the smaller will be 0-2

13 Power of a Test The power of a test is the probability of rejectig H 0 give that a specific alterative is true The power of a test ca be computed as - β. Previous example: the probability of a type II error is give by β 0.866, thus the power of the test is The power is a more succict measure of how sesitive the test is for detectig differeces betwee a mea of 68 ad 68.5 To produce a desirable power greater tha 0.8, oe must either icrease α or icrease the sample size 0-3

14 Oe- ad Two-Tailed Tests The ull hypothesis H 0 is always stated usig the equality sig to specify a sigle value easily cotrolled I a hypothesis, the alterative is oe-sided, ad is called a oe-tailed test. H 0 : θ θ 0 ; H : θ > θ 0 Right tail side H 0 : θ θ 0 ; H : θ < θ 0 left tail side I a hypothesis, the alterative is two-sided, ad is called a two-tailed test. H 0 : θ θ 0 ; H : θ θ 0 0-4

15 How are H 0 ad H Chose? Example 0.: A maufacturer of a certai brad of rice cereal claims that the average saturated fat cotet does ot exceed.5 grams. State the ull ad alterative hypotheses to be used i testig the claim ad determie where the critical regio is located. The claim should be rejected oly if is greater tha.5 Oe-tailed test H 0 :.5; H : >.5 Example 0.2: A real estate aget claims that 60% of all private resideces beig built today are 3-bedroom homes. State the ull ad alterative hypotheses to be used i testig the claim ad determie the locatio of the critical regio. The higher or lower test statistic tha 0.6 would reject the claim Two-tailed test H 0 : p 0.6; H : p

16 Approach to Hypothesis Testig with Fixed α. State the ull ad alterative hypotheses 2. Choose a fixed sigificace level. 3. Choose a appropriate test statistic ad establish the critical regio based o. 4. From the computed test statistic, reject H 0 if the test statistic is i the critical regio. Otherwise, do ot reject. 5. Draw scietific or egieerig coclusios. 0-6

17 P-Values for Decisio Makig It had become customary to choose a α of 0.05 or 0.0 ad select the critical regio accordigly. to cotrol the type I error However, this approach does ot accout for values of test statistics that are close to the critical regio A P-value is the lowest level of sigificace at which the observed value of the test statistic is sigificat o fixed α is determied The coclusio is made o the basis of p-value i harmoy with the subject judgmet of the egieer 0-7

18 P-value Approach Sigificat testig approach State ull ad alterative hypotheses. Choose a appropriate test statistic. Compute P-value based o computed value of test statistic. Use judgmet based o P-value ad kowledge of scietific system. 0-8

19 0-9 Tests Cocerig a Sigle Mea Variace Kow / 2 / 2 / 2 / or / regio : Critical / Acceptace regio : /, / : Stadardizatio of :, : Hypothesis : α α α α α z z z z z z P Z H H < > < <

20 Example 0.3 A radom sample of 00 recorded deaths i the Uited States durig the past year showed a average life spa of 7.8 years. Assumig a populatio stadard deviatio of 8.9 years, does this seem to idicate that the mea life spa today is greater tha 70 years? Use a 0.05 level of sigificace. P PZ >

21 Example 0.4 A maufacturer of sports equipmet has developed a ew sythetic fishig lie that he claims has a mea breakig stregth of 8 kilograms with a stadard deviatio of 0.5 kilogram. Test the hypothesis that 8 kilograms agaist the alterative that 8 kilograms if a radom sample of 50 lies is tested ad foud to have a mea breakig stregth of 7.8 kilograms. Use a 0.0 level of sigificace. P P Z > PZ <

22 Relatioship to Cofidece Iterval Estimatio For the case of a sigle populatio with mea ad variace 2 kow, both hypothesis testig ad cofidece iterval estimatio are based o the R.V. Z a 0 zα / 2 b + 0 zα / 2 We have -α 00% cofidece iterval o The testig of H 0 : 0 agaist H 0 : 0 at a sigificace level α ad rejectig H 0 if 0 is ot iside the cofidece iterval 0-22

23 0-23 Choice of Sample Size The sample size is usually made to achieve good power for a fixed α ad fixed specific alterative. Suppose that we wish to test the hypothesis: H 0 : 0, H : > 0 with a sigificace level α For a specific alterative, say 0 +δ, the power of the test is whe 0 δ β + > a P < + < + + < z Z P a P a P δ δ δ δ β α whe a z α 0 δ α β z z We ca choose the sample size δ z α z β +