Question

according to the institute for college access and success, 80% of student loans are from the federal government. a financial aid officer at uwec suspects that uwecs percentage of federal loans is higher than this. in other words, they want to test (h_0:pi = 0.80) (h_a:pi>0.80) a. the financial aid officer starts by taking a random sample of 5 student loans held by uwec students, and finds that all 5 are federal loans. use the multiplication rule for independent events to find the p - value. 32768 the financial aid officer decides to increase the sample size to 80, and finds 68 federal loans. b. use the binomial distribution to find the p - value. c. use a normal approximation to find the p - value. 0.131776284

Explanation:

Step1: Recall binomial distribution formula

The binomial probability formula is $P(X = k)=C(n,k)\times\pi^{k}\times(1 - \pi)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $\pi$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. For a one - sided right - tailed test with $H_0:\pi = 0.80$ and $H_a:\pi>0.80$, the p - value is $P(X\geq k)=\sum_{i = k}^{n}C(n,i)\times\pi^{i}\times(1 - \pi)^{n - i}$. Here, $n = 80$ and $k = 68$, $\pi=0.80$.
First, calculate the binomial coefficient $C(n,k)=\frac{n!}{k!(n - k)!}=\frac{80!}{68!(80 - 68)!}=\frac{80!}{68!12!}$.

Step2: Calculate binomial probabilities and sum

$P(X\geq68)=\sum_{i = 68}^{80}\frac{80!}{i!(80 - i)!}\times(0.8)^{i}\times(0.2)^{80 - i}$.
$P(X = 68)=\frac{80!}{68!12!}\times(0.8)^{68}\times(0.2)^{12}$
$P(X = 69)=\frac{80!}{69!11!}\times(0.8)^{69}\times(0.2)^{11}$
$\cdots$
$P(X = 80)=\frac{80!}{80!0!}\times(0.8)^{80}\times(0.2)^{0}$
Using a statistical software or calculator with binomial probability functions (e.g., in R: 1 - pbinom(67,80,0.8)), we find that $P(X\geq68)\approx0.1318$.

Answer:

0.1318