At a new exhibit in the Museum of Science, people are asked to choose between 76

Question

At a new exhibit in the Museum of Science, people are asked to choose between 76 or 169 random draws from a machine. The machine is known to have 98 green balls and 61 red balls. After each draw, the color of the ball is noted and the ball is put back for the next draw. You win a prize if more than 69% of the draws result in a green ball. Use Table 1. a. Calculate the probability of getting more than 69% green balls. (Round “z” value to 2 decimal places, and final answer to 4 decimal places.) n Probability 76 169 b. Would you choose 76 or 169 draws for the game? 76 balls 169 balls

Explanation / Answer

Solution:

From the given information we have the proportion of green balls is,

p = 98/ 98+61 = 0.6163
a) Compute the probability of getting more than 69% green balls. That is P(p > 0.69)
For sample size n = 76

P(p > 0.69) = P(p-p/sqrt(p(1-p)/n) > 0.69- 0.6163/sqrt(0.6163(1-0.6163)/76)
= P(Z > 1.32)
= 1-P(Z 1.32)
= 1- 0.9066
= 0.0934
For sample size n = 169
P(p > 0.69) = P(p-p/sqrt(p(1-p)/n) > 0.69- 0.6163/sqrt(0.6163(1-0.6163)/169)
= P(Z > 1.97)
= 1-P(Z 1.97)
= 1- 0.9756
= 0.0244
b) Since the probability value is high when the sample size is 76, we would choose 76 draws for the game.

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