King, a golden retriever, can find a tennis ball hidden in his yard 89% of the time, while Tessa, a Labrador mix, can find a tennis ball hidden in her yard 94% of the time. Suppose King’s owner hides a tennis ball in his yard 100 times and Tessa’s owner hides a tennis ball in her yard 150 times. Let K = the proportion of times King finds the tennis ball and T = the proportion of times Tessa finds the tennis ball. What is the probability that the proportion of tennis balls King finds is greater than the proportion Tessa finds?Find the z-table here.
The difference in sample proportions, K − T, has mean 0.89 − 0.94 = −0.05 and standard error √[(0.89)(0.11)/100 + (0.94)(0.06)/150] ≈ 0.0368. Thus, a difference of 0 corresponds to z = [0 − (−0.05)]/0.0368 ≈ 1.36. The probability that K − T is greater than 0 is the upper-tail probability beyond z = 1.36, about 0.088, so the answer is 0.088.
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