Question

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 O 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 of this test versus the alternative hypothesis that p = 0.075?

Answer 1

Please refer below images for solution.

Solution : final answers in the stipulated format are given below. Back-up theory and details of calculations follow at the end. part (a): 1.96 Answer 1 Critical value = part (6) conclusion since 12 cal! 2 Zenit, - or equivalently, since p-value > a, Ho is accepted. There is enough evidence to suggest that the game is fair. Hence the game is is fais. Answer a

part (2) ß= p (Type 11 Error) = probability of accepting nell hypothesis when it is not true i.e Alternative is true. Now, we reject the well hypothesis when 121 = IC Phar-Pol SpoC1-P)/n) />1.96.so we accept the nell hypothesis when 121= 1 Phat - Po) / JPO CI-Poln) 151.96 i el (Phat -0.) | J Co.1 x 0.9148) | or I Phat) 50 1+1.96560.1X0:9/48] s1.96 or I Phat I 0.1849. Now under the alternate hypothesis, p=0.075 sof Phat -0.015/10.075 80.925/48) ~ N(0.1) Hence P[I Phati] = 0.1849) - p [(2)] = (0.1849 -0.0751 500.00 = P(izi) < 4-6663 - 0.000016 [using excel function: statical NORMS DIST This ß= 0.000016 Answel

Back-up theory and details of calculations Let X= number of times out of 582 times when the first number was zero. Then, x n B.(np) where n= Sample size p=probability that 1st meestel was o. Hypotheses: Null Hoop=Po = 1/10 or olv Alternative HAP 0-1 Test static 2= (Phat - PolJP C1 - Poln) Phat - sample proportion n= sample size. calculations where Po 0.10 582 48 Х Phat 0.0825 1.4093 Leal 0.05 a 1.9600 Legit p-value / 0.1587

p-value Distribution, significance level, a critical value and cender Ho, distribution of a can be by standard normal distribution, provided. appronionate np. and no cI-P.) are, both greatee than 10. a level of significance of 27., critical Value = N(o,). and uppee (0/2)% of p-value = P(Z > 12 call) Using Excel function: Statistical {NORMSINU found as and NORMSDIST these are So, given shown is is the table Decision : Since Izcall< Zeit, or equivalently, since p-value xa Ho is accepted conclusion that Suggest There is enough evidence to is fair. the game