Question

Hint: Critical region is a synonym for rejection region. In a fair “Pick 3” game, the first digit drawn should be a zero with probability 0.1 because there are ten equally likely outcomes. A lottery player suspects that the game is unfair. To test his theory, he records the numbers drawn over two years which constitute 582 plays. a) (4 points) Write down the critical region for testing whether or not the probability of drawing a 0 first is actually 1/10. Use a level of a = 0.05. b) (7 points) If he finds that 48 out of 582 times the first number was zero, what does the test conclude? Use both RR and p-value to reach the conclusion. c) (4 points) What is the ß of this test versus the alternative hypothesis that p= 0.075?

Answer 1

(b) from the observed data critical value 1.96 conclusion: since I callezerit , or equivalently, since p-value &a, accepled. Ho There enough evidence to suggest that the game is bail Hence conclude that there is enough evidence suggest that the game bair (C) B - hypothesis when Now, the we pl Type û error )= probability of accepting a rull it is not true, vie., Alternative is. frue. reject nell hypothesis when 12) = || Prat -po)l(Poll-80) 1 >1.96, so the null hypoth thesis when (2) Phat -Pol Poli-pa) lie., I Phant 01 / 10:1x0.9 - accept we 21-967 €1.96 48

Now the 31 49 or (Prat) <0,1 + 1.96 0.1*0.9 48 or. I phat < 0.1949. undee alternative hypothesis ,p=0.075, so I(Phat -0.0751000 100-015 x0-945 No.1) Hence P{Phatt <0. 1849 P[1213(0.1849 -0.075/50-07) P (Izle 4.6663) 0.000016 ( using Excel buns's : Statistical RM BEST) This ß 0.0ooolt Backup Theory and Details of calculations out of 582 times when the first number cet xa NORM DIST No of times zelo where A = sample size and pa probability number was zelo us 10 Then N B (nip) that the 1st Hypothesis in Null Ho! P= Post Or 0.1 Allarnative : Ha:P 0.1 Test Statistic: (Phat apo) Poli-po) 2 = where Prata Sauple proportion na sample size n Po n Phat Ical. Zerit p-value 010 582 48 0.0825 -1.4093 0.05 1.96000.1587

by are (%23% do Distribution, significance level ed ecritical a value & p-value undee Ho, distribution of 2 can be approximated Standard normal distribution, prouided npo and apoll-Po) both greater than 10. a level of significance of a yo critical value supper So, given Nail and privalue =p(23/2 call) EXCEL function statistical NORMSINU and NORMSDIS T thease found Shown the above table. Decision :- since la calle 2 crit, or equivalently or equivalently since p-value sa, accepted. conclusion in enough euidence to suggest that the game bail - using ave Ho le There is is hope this answer will helpful for you please give up- vote for my hard work