Question

Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe as the random variable Y10 = x and note that Yio follows a Binomial distribution: Y1o Bin(10,1/2) ال » ال ال ال ال ا ال ا

Answer 1

Here, XiSi, if coin frip i result in heads lo , if coin trip i result in tails Now flip the coin n=10 times. * Let us define total no. of heads observed as the random variable, Yo ..[ xin Clearly Yo follows a Binomial Distribution. Ye ~ Bin (n=10, p= ) The pmf is, + ( 4 ) : ( 3 ) * (- "* 8 - 12% 0.w. Opland, en is natural number. The cumulative prob. that we will observe a total of O heads in 10 flips , (10-0) PC Yo so]: PC Yoo go ] 1.1.(1) . (6) The cumulative boob, that we will observe a total of I heads 10 flips . P[ Yos!] =P [ Yo =0] + P ( Yo17 - (7)" + (*10). (1) (1) "0" ; (f) + 10 x ( ) (1) * (16+1) = 11% ()" in

cumulative The prob. that, we will observe a total of a heads in 10 flips, P[ Yio ??] =P [ Y1031] + P [ Yoo =2] = 1X (1)' + () (+)2 (+- 1 ) *-* = n* (47'()" = (4) ( 11 + 5x9) z (2)'0 .x 56 llo Now we want to test . Ho: p = 8 h against Hy ; p < Ži. a Note that under Ho : P = 2 , Yion Bin (n=10, p = 1/2 ). distri bution which is completely specified. to as a test statistic to So , Yio can be used compute p-values. The observed value of %10 is a , which is given. ise. Yo (observed ) when Hii p < ², 2. it Now, is expected that Y10 should be smaller than 2. Hence, p value = Pro [ Yo < 2] 0.0546875 Now, Let 0-0.05 be the chosen level of significance, Decision If It rule: when a a is is the the chosen chosen level of level of significance, brame {a ; we reject Ho, at level a p value >d ; we o fail to reject Ho, at level a.

Here , p value = 0.054 6875 > 2:0.05 So, we fail to reject in the null hypothesis Ho : p = 1 at level d =0.05 Alternatively conversely, we say that there is not enough evidence against the null hypothesis Ho that the coin is fair at level a=0.06. ie. we will fail to reject the null hypothesis with tonfiel 95% confidence. So, we believe that Ho holds with 95% confidence.