Two brothers named Mario and Luigi like to compete against each other by playing

Question

Two brothers named Mario and Luigi like to compete against each other by playing a certain video game. One afternoon, Mario wins and is a being a jerk about it. Luigi is convinced that he is better than Mario and starts keeping track of who wins every time they play. Over the next 40 games, Luigi wins 22 of them (55%). Luigi knows that 22 wins in 40 games is not enough evidence to conclude that the brothers are not evenly matched. Assuming Luigi continues to win exactly 55% of the games that they play, and that he tests the null model after every 10 games, how many games will it take for Luigi to have sufficient evidence that the two brothers are not evenly matched?

Explanation / Answer

here null hypothesis: Ho: p=0.5 ( both are on equal level)

alternate hypothesis: Ha: p >0.50 (Luigi is better player)

for 0.05 level ;critical value z =1.645

for std error =sqrt(pq/n) =sqrt(0.5*0.5/n) =0.5/sqrt(n)

hence for suffiicient evidence of being a better player

test statisitc z >1.645

(phat-p)/std error >1.645

(0.55-0.50)/(0.5/sqrt(n)) >1.645

sqrt(n) >16.45

n >= 270.6 games

hence Luigi must play more than 270 games or to check at n=280 games to have sufficient evidencethat he is better player,

( please revert for any clarification required)

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