Question

.. We are interested in testing whether or not a coin is balanced based on the number of heads Y on 36 tosses of the coin (Ho : p = 0.5 versus HA : P = 0.5). Suppose we use the rejection region {y: ly - 181 > 4}. (a) In terms of this problem what would be a type I and type II error? (b) Find the level, a, of the test. (Hint: Is a Normal approximation appropriate?) (c) Find the value of B if p = 0.4. (Hint: Is a Normal approximation appropriate?) (d) Suppose after flipping 36 times we observe 15 heads. Based on the given rejection region, what is our conclusion?

Answer 1

grejection eregion is y-181-4 Ho : p=0.5 Hi P805 y = number of heads in n. 36 tosses Ely) anpir (y) = npg ca) Type 1 error Plreject Hol Ho is true ) in this problem it is the probablity that number It heads greater than 22 and number of heads less than it when probablity of getting a head is o's Typeí z exto x = caccept Ho lh, is true Jay-D Yeject HolHiis true) In this problem is the probablity that number of heads 14 4 22 when probablity of getting a head is not equal to os bo az PCIV-18/>411) ? PC4218 +41H0)+ PCV 418-4140 papy-np hP9 214-np Inpa --PC27 22-3605) Inpa 360.5o. + PZ z 14-36005) 36(0J)ois 22 np jupg) + py-ne

=P(Z >1:33) + P(Z <-1:33) = 0.0917 +0.0917 0.1834 (C) B Type 2 error = -placcept Holp=04) = 1-0 (reject Holp=0.4) -1-p (y-181>4|p 2014) 31-17 C4>22(p =04)-pcy_1412=0 4) - 1-PCY72210304)-PCY-1472-04) 31 - p (2722-36 (0.4) 36 (0.450.6 -Plze 14 -36 (0.4) (300:40:6 -1-PC2>2.58)-PC2C-0.136) | -0.005-0.446 20.549 - 1-PCV >2217 -0.4) -p (4 <1412=0-4) - 1-PC2->22-36 (0.4) -P2¿14-36 (0-4) 3600.4J0:6 P1-PC2 >2.58-PC26-0.136) 136 (04)06 -| -0.005 -0.46 -0.549 We hare used normal approximation as we

Know that n is large greater than 30) d) since the rule is rejeet Hoit y 222 ory<14 hence we have y=is and we don't have enough evidence to reject in this case,