Respuesta :
Answer:
Probability that 300 or more will be recycled is 0.0314.
Step-by-step explanation:
We are given that one environmental group did a study of recycling habits in a California community. It found that 73% of the aluminum cans sold in the area were recycled.
Also, 388 cans are sold today.
The above situation can be represented through Binomial distribution;
[tex]P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....[/tex]
where, n = number of trials (samples) taken = 388 cans
r = number of success = 300 or more
p = probability of success which in our question is % of aluminum
cans sold in the area that were recycled, i.e; 73%
LET X = Number of cans that are recycled
SO, X ~ Binom(n = 388, p = 0.73)
Now, here we can't find the probability that 300 or more will be recycled using binomial distribution because the sample size is very large here (n > 30), so we will use Normal approximation to find the respective probability.
So, mean of binomial distribution = E(X) = [tex]n\times p[/tex]
= [tex]388 \times 0.73[/tex] = 283.24
Standard deviation of binomial distribution = S.D.(X) = [tex]\sqrt{n\times p \times (1-p)}[/tex]
= [tex]\sqrt{388 \times 0.73 \times 0.27 }[/tex]
= 8.74
Let Y = Number of cans that are recycled for normal approximation;
So, Y ~ Normal([tex]\mu=283.24,\sigma = 8.74[/tex])
The z-score probability distribution for normal distribution is given by;
Z = [tex]\frac{Y-\mu}{\sigma}[/tex] ~ N(0,1)
Now, probability that 300 or more will be recycled is given by = P(Y [tex]\geq[/tex] 300) = P(Y > 299.5) --------------{using continuity correction}
P(Y > 299.5) = P( [tex]\frac{Y-\mu}{\sigma}[/tex] > [tex]\frac{299.5-283.24}{8.74}[/tex] ) = P(Z > 1.86) = 1 - P(Z [tex]\leq[/tex] 1.86)
= 1 - 0.9686 = 0.0314
The above probability is calculated using z table by looking value of x = 1.86 in z-table.
Therefore, the probability that 300 or more will be recycled is 0.0314.
