Question

1. a certain midwestern university claims that 90% of their football players complete their degree in 5 years. the ncaa investigates this claim by selecting a random sample of 20 football players that have been part of the program within the past 5 years. 14 of these players received their degree, while 6 did not receive their degree. if you were the investigator, what would you conclude about the university’s claim? explain your reasoning with probability.

Explanation:

Step1: Calculate the expected number of players who complete degree

If the claim of \(90\%\) is true, for a sample of \(n = 20\) players, the expected number of players who complete their degree is \(E=np\), where \(p = 0.9\). So \(E=20\times0.9=18\).

Step2: Analyze the probability of getting the observed result

We can use the binomial probability formula \(P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\). Here \(n = 20\), \(p=0.9\), \(k = 14\). \(\binom{20}{14}=\frac{20!}{14!(20 - 14)!}=\frac{20\times19\times18\times17\times16\times15}{6\times5\times4\times3\times2\times1}=38760\). \(P(X = 14)=\binom{20}{14}(0.9)^{14}(0.1)^{6}\)

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A probability of approximately \(0.0088\) (or \(0.88\%\)) is very low. In hypothesis - testing (using probability as a guide), if the probability of observing a result (or a more extreme result) under the null hypothesis (the university's claim is true) is very low (commonly, if \(P<0.05\)), we reject the null hypothesis.

Answer:

The university's claim that \(90\%\) of their football players complete their degree in 5 years is likely false. The probability of having only 14 out of 20 players complete their degree (if the \(90\%\) success - rate claim were true) is approximately \(0.88\%\), which is a very low probability. So, based on probability reasoning, we would reject the university's claim.