Question 1. A manufacturer produces and sells a product X for $500 per unit. Two facilities are available for production.

- Old facility: has old machinery and a daily yield of product X that follows the normal distribution with μ₁ = 10, σ₁² = 10.
- New facility: has new machinery and a daily yield of product X that follows the normal distribution with μ₂ = 20, σ₂² = 5.

The yield of the two facilities is independent of one another. Answer the following:

(a) What is the distribution of the total daily yield?

(b) Management is considering renovating the old facility to have a yield similar to the new one. If the daily demand on the old facility for the next year is fixed at 15 units per day, does the expected lost profit justify an investment of $100,000?

(c) If management incurs a fixed penalty of $500 per day for backlogs, What is the exact probability that the back log cost for the next 2 week is 15000 if the daily demand is normally distributed with μ_d = 25, σ_d² = 7?

(d) Approximate the probability that the backlog cost in the next year does not exceed $15,000?

Question 2. The Rayleigh distribution has the following probability density function:

f(x) = x/θ.e-^_x2_/2θ , x > 0, 0 < θ < ∞

(a) Find the second moment of the distribution above.

(b) Using the result in (a), construct an unbiased estimator for θ.

(c) Find the maximum likelihood estimator of θ.

(d) What is the MSE of the maximum likelihood estimator of θ?

(e) Calculate θ^{^}_mle from the following sample:

3.07 2.46 0.61 3.23 0.55 3.35 4.46 2.37 5.17 0.90

(f) Using θ^{^}_mle, an α level confidence interval on θ can be obtain using the following formula:

2n²θ^{^}_mle/Χ²_α⁄2,n-1 ≤ θ ≤ 2n²θ^{^}mle/Χ²_{1-α⁄2, n-1}

Construct a 95% confidence interval on θ using the sample from part (e).

(g) From the confidence interval obtained from part (f), would you reject the hypothesis that θ = 4?

Question 3. The quantity of heat that is required to boil 2 litres of water was measured. The experiment was performed 10 times. The results in (kJ) are as follows:

656, 631, 627, 644, 661, 670, 623, 649, 651, 658

Assuming that the heat follows a normal distribution, answer the following:

(a) If σ = 10, construct a 95% two-sided confidence interval on the mean heat.

(b) If σ is unknown, how does the interval in (a) change?

(c) Is there strong evidence to conclude that the variance exceeds 100? Use α = 0.01.

(d) Find the P-value for the test in (c)?

(e) How can part (c) be answered using a confidence bound on σ²?

(f) How large does the sample need to be in order to obtain a confidence interval with an error in estimating the mean that does not exceed 10? Use σ = 10 and α = 0.01.

(g) If the true mean is 670, what is the type-2 error for the test in (a)?