To solve this problem, we'll use some basic concepts from statistics, specifically the binomial distribution. Here's a step-by-step breakdown:
1. Determine the number of trials (n):
- Here, the number of dentists in your area is [tex]\( n = 120 \)[/tex].
2. Determine the probability of success (p):
- According to the commercial, the probability that a dentist recommends the toothpaste is [tex]\( p = \frac{4}{5} = 0.8 \)[/tex].
3. Calculate the expected number of dentists who recommend the brand (mean [tex]\(\mu_x\)[/tex]):
- The expected number of successes in a binomial distribution is calculated as [tex]\( \mu_x = n \times p \)[/tex].
- Substituting the given values:
[tex]\[
\mu_x = 120 \times 0.8 = 96
\][/tex]
4. Calculate the standard deviation ([tex]\(\sigma\)[/tex]):
- The standard deviation of a binomial distribution can be calculated using the formula [tex]\( \sigma = \sqrt{n \times p \times (1 - p)} \)[/tex].
- Substituting the given values:
[tex]\[
\sigma = \sqrt{120 \times 0.8 \times (1 - 0.8)}
= \sqrt{120 \times 0.8 \times 0.2}
= \sqrt{120 \times 0.16}
= \sqrt{19.2}
\approx 4.38
\][/tex]
So, the expected number of dentists who recommend the brand is [tex]\( 96 \)[/tex], and the standard deviation is approximately [tex]\( 4.38 \)[/tex].
Therefore, the correct option is:
[tex]\[
\mu_x = 96 ; \sigma = 4.38
\][/tex]
