Question

determine whether you can use a normal distribution to approximate the binomial distribution. if you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. if you cannot, explain why and use a binomial distribution to find the indicated probabilities. identify any unusual events. explain.
in a survey of u.s. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re - enroll. you randomly select 39 college students. find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 24, (b) at least 29, and (c) at most 23.
determine whether a normal distribution can be used to approximate the binomial distribution. choose the correct answer.
a. no, because np < 5.
b. yes, because both np≥5 and nq≥5.
c. no, because nq < 5.
(a) find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is exactly 24.
(round to four decimal places as needed.)

Explanation:

Step1: Identify binomial parameters

Let $n = 39$ (number of trials), $p=0.62$, $q = 1 - p=1 - 0.62 = 0.38$.

Step2: Check normal - approximation conditions

$np=39\times0.62 = 24.18\geq5$ and $nq=39\times0.38 = 14.82\geq5$, so we can use normal - approximation. The mean of the normal distribution is $\mu=np = 24.18$ and the standard deviation is $\sigma=\sqrt{npq}=\sqrt{39\times0.62\times0.38}=\sqrt{9.19}\approx3.03$.

Step3: Continuity correction for $P(X = 24)$

For $P(X = 24)$ in binomial, in normal approximation we find $P(23.5\lt X\lt24.5)$. First, standardize the values: $z_1=\frac{23.5 - 24.18}{3.03}=\frac{- 0.68}{3.03}\approx - 0.22$ and $z_2=\frac{24.5 - 24.18}{3.03}=\frac{0.32}{3.03}\approx0.11$.

Step4: Find probabilities using z - table

$P(-0.22\lt Z\lt0.11)=\Phi(0.11)-\Phi(-0.22)$. From the standard - normal table, $\Phi(0.11) = 0.5438$ and $\Phi(-0.22)=0.4129$. So $P(-0.22\lt Z\lt0.11)=0.5438 - 0.4129 = 0.1309$.

Answer:

0.1309