Question

Making handcrafted pottery generally takes two major steps: wheel throwing and firing. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40 minutes and 60 minutes and standard deviations of 2 minutes and 3 minutes, respectively. Assume the time of wheel throwing and time of firing are independent random variables. (d) What is the probability that a piece of pottery will be finished within 95 minutes

Answer 1

Answer:

The probability that a piece of pottery will be finished within 95 minutes is 0.0823.

Step-by-step explanation:

We are given that the time of wheel throwing and the time of firing are normally distributed variables with means of 40 minutes and 60 minutes and standard deviations of 2 minutes and 3 minutes, respectively.

Let X = time of wheel throwing

So, X ~ Normal($\mu_x=40 \text{ min}, \sigma^{2}_x = 2^{2} \text{ min}$)

where, $\mu_x$ = mean time of wheel throwing

            $\sigma_x$ = standard deviation of wheel throwing

Similarly, let Y = time of firing

So, Y ~ Normal($\mu_y=60 \text{ min}, \sigma^{2}_y = 3^{2} \text{ min}$)

where, $\mu_y$ = mean time of firing

            $\sigma_y$ = standard deviation of firing

Now, let P = a random variable that involves both the steps of throwing and firing of wheel

SO, P = X + Y

Mean of P, E(P) = E(X) + E(Y)

                   $\mu_p=\mu_x+\mu_y$

                        = 40 + 60 = 100 minutes

Variance of P, V(P) = V(X + Y)

                               = V(X) + V(Y) - Cov(X,Y)

                               = $2^{2} +3^{2}-0$  

{Here Cov(X,Y) = 0 because the time of wheel throwing and time of firing are independent random variables}

SO, V(P) = 4 + 9 = 13

which means Standard deviation(P), $\sigma_p$ = $\sqrt{13}$

Hence, P ~ Normal($\mu_p=100, \sigma_p^{2} = (\sqrt{13})^{2}$)

The z-score probability distribution of the normal distribution is given by;

                           Z  =  $\frac{P- \mu_p}{\sigma_p}$  ~ N(0,1)

where, $\mu_p$ = mean time in making pottery = 100 minutes

           $\sigma_p$ = standard deviation = $\sqrt{13}$ minutes

Now, the probability that a piece of pottery will be finished within 95 minutes is given by = P(P $\leq$ 95 min)

     P(P $\leq$ 95 min) = P( $\frac{P- \mu_p}{\sigma_p}$ $\leq$ $\frac{95-100}{\sqrt{13} }$ ) = P(Z $\leq$ -1.39) = 1 - P(Z < 1.39)

                                                            = 1 - 0.9177 = 0.0823

The above probability is calculated by looking at the value of x = 1.39 in the z table which has an area of 0.9177.