Question

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. (Use the normal approximation. Round your answers to four decimal places.) (a) If 388 cans are sold today, what is the probability that 300 or more will be recycled

Answer 1

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;

$P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....$

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) =  $n\times p$

                                                        = $388 \times 0.73$ = 283.24

Standard deviation of binomial distribution = S.D.(X) = $\sqrt{n\times p \times (1-p)}$

                                                                        = $\sqrt{388 \times 0.73 \times 0.27 }$  

                                                                        = 8.74

Let Y = Number of cans that are recycled for normal approximation;

So, Y ~ Normal($\mu=283.24,\sigma = 8.74$)

The z-score probability distribution for normal distribution is given by;

                  Z = $\frac{Y-\mu}{\sigma}$  ~ N(0,1)

Now, probability that 300 or more will be recycled is given by = P(Y $\geq$ 300) = P(Y > 299.5)      --------------{using continuity correction}

        P(Y > 299.5) = P( $\frac{Y-\mu}{\sigma}$ > $\frac{299.5-283.24}{8.74}$ ) = P(Z > 1.86) = 1 - P(Z $\leq$ 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.