One of your employees has suggested that your company develop a new product. A survey is designed to study whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating definitely would not purchase, · · ·, and 5 indicating definitely would purchase. For an initial analysis, you will record the responses 1, 2, and 3 as No, and 4 and 5 as Yes.
a. 5 people are surveyed. What is the probability that at least 3 of them answered Yes?
b. 100 people are surveyed. What is the approximate probability that between 45% to 52% of people answered Yes?
For part a) There are 5 choices that people can respond by 1, 2 ,3 ,4 and 5. Since 1, 2 and 3 are considered "No", the probability of someone answering "No" is 3/5. For choices 4 and 4, the probability of someone responding with that is 2/5.
This looks like it fallows a binomial distribution so I calculated the probability of P(3) + P(4) + P(5).
However for part b), I'm confused. I can calculate the probability of someone saying yes but I don't know how to calculate the probability that a percentage of people saying yes. Does anyone know how to approach this question? I though about using the Z table, but that already calculates area.
Let $X$ denote the number of people answering yes. Then $X$ is a bionomial random variable.
Let $n$ denote the number of trials (i.e. people surveyed). Let $p$ denote the probability of a yes-response on a given trial, and let $q$ denote the probability of a no-response. Then, as you've argued, we have $$ p = 2/5, \qquad q = 3/5. $$ Then in part (a), the probability of at least three people answering yes is $$ P(x = 3) + P(x = 4) + P(x=5) = \ldots. $$ So far so good!
Now for part (b):
You see, $45%$ of $1000$ is $450$, and $52%$ of $1000$ is $520$. So I reckon the probability you want is $$ \begin{align} P(X = 450) + P(X = 451) + \cdots + P(X = 520) &= \sum_{r = 450}^{520} P(X = r) \\ &= \sum_{r = 450}^{520} {1000 \choose r} \left( \frac{2}{5} \right)^r \left( \frac{3}{5} \right)^{1000 - r } \\ &= \cdots \end{align} $$