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Answer:
The probability that 85% or more of the sampled mussels will be infected is 0.1057.
Step-by-step explanation:
Let X = number of mussels infected with an intestinal parasite.
The probability that a random selected mussel is infected with an intestinal parasite is, p = 0.80.
A random sample of n = 100 mussels from the population are examined by a marine biologist.
The random variable X follows a Binomial distribution with parameters n = 100 and p = 0.80.
But the sample selected is too large, i.e. n = 100 > 30.
So a Normal approximation to binomial can be applied to approximate the distribution of [tex]\hat p[/tex], the sample proportion of mussels infected with an intestinal parasite, if the following conditions are satisfied:
- np ≥ 10
- n(1 - p) ≥ 10
Check the conditions as follows:
n × p = 100 × 0.80 = 80 > 10
n × (1 - p) = 100 × (1 - 0.80) = 20 > 10
Thus, a Normal approximation to binomial can be applied.
So, the distribution of [tex]\hat p[/tex] is:
[tex]\hat p\sim N(p, \frac{p(1-p)}{n} )[/tex]
Compute the probability that 85% or more of the sampled mussels will be infected as follows:
Apply continuity correction:
[tex]P(\hat p\geq 0.85)=P(\hat p>0.85+0.50)[/tex]
[tex]=P(\hat p>0.90)\\[/tex]
[tex]=P(\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}>\frac{0.85-0.80}{\sqrt{\frac{0.80(1-0.80)}{100}}})[/tex]
[tex]=P(Z>1.25)\\=1-P(Z<1.25)\\=1-0.89435\\=0.10565\\\approx0.1057[/tex]
*Use a z-table for the probability.
Thus, the probability that 85% or more of the sampled mussels will be infected is 0.1057.
The probability that 85% or more of the sampled mussels will be infected would be "0.1303".
Probability
According to the question,
Sample number, n = 100
Number of infected individuals, p = 80% or,
= 0.8
Now, μ = np
= 100 × 0.8
= 80
and, σ = √np (1 - p)
= √100 (0.8) (0.2)
= 4
The z-value will be:
→ z = [tex]\frac{x- \mu}{\sigma}[/tex]
= [tex]\frac{84.5 - 80}{4}[/tex]
= 1.125
hence, the probability be:
p (z > 1.125) = 1 - p (z [tex]\leq[/tex] 1.125)
= 1 - 0.8697
= 0.1303
Thus the above response is correct.
Find out more information about probability here:
https://brainly.com/question/24756209
