Question

(1 point) Use normal approximation to estimate the probability of passing a true/false test of 70 questions if the minimum passing grade is 80% and all responses are random guesses.

Answer 1

Since , the questions are true/false type.

Now , there are only two possibilities.

Therefore , the probability of the correct answer is 0.5

Since , the minimum passing grade is 80%

Here , by using the normal approximation ,

Therefore , the mean and standard deviation is ,

$\mu_{\hat{p}}=p=0.5$

$\sigma_{\hat{p}}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.5(1-0.5)}{70}}=0.0598$

Now ,

$P(P\geq 0.80)=P(\frac{p-\mu_{\hat{p}}}{\sigma_{\hat{p}}}\geq \frac{0.80-0.5}{0.0598})=P(Z\geq 5.02)$

$=1-P(Z<5.02)=1-\Phi(5.02)=1-1\approx 0$ ; From standard normal distribution table

Therefore , the probability of passing is approximately 0