Question

A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 50% of this population prefers the color green. If 15 buyers are randomly selected, what is the probability that exactly a third of the buyers would prefer green?

Answer 1

Answer:

0.09

Step-by-step explanation:

Given that 50% of this population prefers the color green.

Let p the probability that one person selected from the population prefer the green color of the car. So,

p=0.05

There is only two chance, any person either prefer the green color or not, assuming this holds true for every person, so the mentioned population can be assumed as Bernoulli's population.

By using Bernoulli's theorem, the probability of exactly r success of n randomly selected from the Bernoulli's population is

$P(r)=\binom{n}{r}p^{r}{(1-p)}^{n-r}\cdots(i)$

Here, 15 buyers are randomly selected, so, n= 15 and

$r= \frac1 3 \times 15=5$

So, by using equation (i), the probability that exactly 5 buyers would prefer green out of 15 randomly selected buyers is

$P(r=5)=\binom{15}{5}(0.5)^{5}{(1-0.5)}^{15-5}$

$=\binom{15}{5}(0.5)^{5}{0.5}^{10}$

$=\binom{15}{5}(0.5)^{15}$

=0.0916

Hence, the probability that exactly 5 buyers would prefer green out of 15 randomly selected buyers is 0.09.