A tennis player usually makes a successful first serve 74% of the time.She buys a new racket hoping that it will improve her success rate.During the first month of playing with her new racket she makes 402 successful first serves out of 500.Is this evidence that with the new racket her success rate has improved? In other words,is this an unusual result for her? Explain.

A) No; we would normally expect her to make 370 first serves with a standard deviation of 19.24.402 is 1.7 standard deviations above the expected value.That's not an unusual result.
B) No; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's not an unusual result.
C) No; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's not an unusual result.
D) Yes; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's an unusual result.
E) Yes; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's an unusual result.

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Joseph Delos · Accepted Answer

Final Answer : E, Explanation :  The expected number of successful first serves is 370 (74% of 500). The standard deviation can be calculated using the formula for the standard deviation of a binomial distribution, which is sqrt(np(1-p)), where n is the number of trials (500) and p is the probability of success (0.74). This calculation gives a standard deviation close to 9.81. A result of 402 successful serves is 3.3 standard deviations above the expected value (32 serves more than expected), which is statistically significant and indicates an unusual result, suggesting her success rate has improved with the new racket.

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