Respuesta :
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([tex]\mu_x=40 \text{ min}, \sigma^{2}_x = 2^{2} \text{ min}[/tex])
where, [tex]\mu_x[/tex] = mean time of wheel throwing
[tex]\sigma_x[/tex] = standard deviation of wheel throwing
Similarly, let Y = time of firing
So, Y ~ Normal([tex]\mu_y=60 \text{ min}, \sigma^{2}_y = 3^{2} \text{ min}[/tex])
where, [tex]\mu_y[/tex] = mean time of firing
[tex]\sigma_y[/tex] = 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)
[tex]\mu_p=\mu_x+\mu_y[/tex]
= 40 + 60 = 100 minutes
Variance of P, V(P) = V(X + Y)
= V(X) + V(Y) - Cov(X,Y)
= [tex]2^{2} +3^{2}-0[/tex]
{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), [tex]\sigma_p[/tex] = [tex]\sqrt{13}[/tex]
Hence, P ~ Normal([tex]\mu_p=100, \sigma_p^{2} = (\sqrt{13})^{2}[/tex])
The z-score probability distribution of the normal distribution is given by;
Z = [tex]\frac{P- \mu_p}{\sigma_p}[/tex] ~ N(0,1)
where, [tex]\mu_p[/tex] = mean time in making pottery = 100 minutes
[tex]\sigma_p[/tex] = standard deviation = [tex]\sqrt{13}[/tex] minutes
Now, the probability that a piece of pottery will be finished within 95 minutes is given by = P(P [tex]\leq[/tex] 95 min)
P(P [tex]\leq[/tex] 95 min) = P( [tex]\frac{P- \mu_p}{\sigma_p}[/tex] [tex]\leq[/tex] [tex]\frac{95-100}{\sqrt{13} }[/tex] ) = P(Z [tex]\leq[/tex] -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.
