Question

a researcher wishes to conduct a study of the color preferences of new car buyers. suppose that 40% of this population prefers the color blue. if 14 buyers are randomly selected, what is the probability that exactly 8 buyers would prefer blue? round your answer to four decimal places.

Answer 1

The probability that exactly 8 out of 14 randomly selected new car buyers prefer the color blue is approximately 0.2361, based on a population where 40% favor the color blue.

This scenario follows a binomial distribution, where each buyer either prefers blue (success) or does not. The probability of success (p), i.e., a buyer preferring blue, is 0.4.

The probability of exactly 8 buyers preferring blue out of 14 can be calculated using the binomial probability formula:

`\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^(n - k) \]`

where:

- n is the number of trials (buyers),

- k is the number of successes (buyers preferring blue),

- p is the probability of success (probability of preferring blue), and

-
`\( \binom{n}{k} \)` is the binomial coefficient, calculated as
`\( (n!)/(k!(n-k)!) \)`.

In this case:

`\[ P(X = 8) = \binom{14}{8} \cdot (0.4)^8 \cdot (0.6)^6 \]`

Calculating this expression will give the probability. Rounding to four decimal places:

`\[ P(X = 8) \approx 0.2361 \]`

So, the probability that exactly 8 out of 14 buyers prefer the color blue is approximately 0.2361.