Question

1. nick tells his sister penny that he gets up when his alarm clock rings the first time (without pressing snooze) about 85% of the time. within the next month, penny takes a random sample of seven days and finds that on four of the days, nick gets up when his alarm clock rings the first time, but on the other three days, he continues to sleep after his alarm clock rings.

a. does penny have some evidence that nick is lying? explain.
b. does penny have proof that nick is lying? explain.

Explanation:

• For part a:
• If Nick's claim of getting up \(85\%\) (or \(p = 0.85\)) of the time when the alarm rings the first - time is true, and we consider a binomial distribution \(X\sim B(n = 7,p=0.85)\) (where \(n\) is the number of trials - days in the sample).
• The probability mass function of a binomial distribution is \(P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\).
• \(P(X = 4)=\binom{7}{4}(0.85)^{4}(1 - 0.85)^{7 - 4}=\frac{7!}{4!(7 - 4)!}\times(0.85)^{4}\times(0.15)^{3}\)
• \(\binom{7}{4}=\frac{7!}{4!3!}=\frac{7\times6\times5}{3\times2\times1}=35\)
• \(P(X = 4)=35\times0.52200625\times0.003375\approx0.0616\)
• A probability of approximately \(6.16\%\) (less than a common significance level of \(10\%\)) for getting exactly \(4\) successes (days of getting up on the first alarm) in \(7\) trials suggests that the observed result is relatively unlikely if Nick's claim is true. So, Penny has some evidence (based on the low probability of the observed sample result under Nick's claimed probability) that Nick is lying.
• For part b:
• Just because the sample result (getting up \(4\) out of \(7\) days) has a low probability under Nick's claimed \(p = 0.85\) does not constitute proof. There is still a non - zero probability (\(P(X = 4)\approx0.0616\)) that such a sample could occur if Nick's claim is true. Also, the sample size \(n = 7\) is very small. A larger sample would be needed to be more confident (in a statistical sense) of proving or disproving Nick's claim.

Answer:

a. Yes. If Nick's claim (\(p = 0.85\)) were true, using the binomial distribution \(P(X = 4)\approx0.0616\) (a relatively low probability for the observed sample result), so there is some evidence he is lying.
b. No. The sample size (\(n = 7\)) is small, and there is still a non - zero probability (\(P(X = 4)\approx0.0616\)) that the sample result could occur if his claim is true.