# Statistics Assignment

Midterm Statistical Inference II, Dr. Jinwook Lee

Student Name:

Problem 1. A sample of 36 sales receipts from an Apple store has the sample mean of $1200. Assume that the population standard deviation is known as $240. Use these values to test whether or not the mean of the sales is different from $1260. Use the significance level α = 0.05.

(a) Find 95% Confidence Interval for a mean.

(b) Hypothesis Testing

– State the null and alternative hypotheses.

– State Decision Rules for both p-value and critical value methods

– Find Test Statistic

– Conclusion

Problem 2. Consider two securities, the first having µ1 = 1 and σ1 = 0.1, and the second having µ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated, with ρ = −0.8.

(a) If you could invest in one security, which one would you choose, and why?

(b) Suppose you invest 50% of your money in each of the two. What is your expected return and what is your risk?

(c) Suppose you invest 80% of your money in security 1 and 20% in security 2. What is your expected return and what is your risk?

Repeat the above three parts with ρ = 0.5.

(d) What are those values? What do you think when comparing these two different correlated cases?

Problem 3. Starbucks manager at LeBow wants to find:

(a) the probability of the total average cost less than or equal to $1000 per day. He is only concerned about the cost of waiting time of customers. Customers spend time at the checkout register with mean of 4 minutes and standard deviation of 4 minutes. On average, he got 100 customers per day. 1 minute of waiting time is considered as cost of $0.5.

(b) the probability of the total expected revenue greater than or equal to $1500 per day. On average, each customer spends $4 at a time with standard deviation of $1.

Problem 4. The exponential p.d.f. is f(x|λ) = λe−λx for x ≥ 0 and λ > 0, and E(X) = λ−1. The c.d.f. is F(x) = P(X ≤ x) = 1 − e−λx. Four observations are made by an instrument that reports x1 = 5,x2 = 3, but x3 and x4 are too large for the instrument to measure and it reports only that x3 > 10 and x4 > 10. (The largest value the instrument can measure is 10.0)

(a) What is the likelihood function?

(b) Find MLE of λ.

Problem 5. Assume that we fit a Poisson distribution to the following accident-frequency data:

Number of accidents (k) Number of weeks (nk) log nk log(k!) log(nk k!/N) 0 38 3.64 0 -0.69 1 26 3.26 0 -1.07 2 8 2.08 0.69 -1.56 3 2 0.69 1.79 -1.85 4 1 0 3.18 -1.15 12 1 0 19.99 15.66

What is the MLE of λ (i.e., the Poisson parameter)? N is the sum of nk (i.e., N = 76) and (as you know)

the Poisson p.m.f is p(i) = e−λ λ i

i! for i = 0, 1, . . . , where λ > 0.

Problem 6. Suppose that an i.i.d. sample of size 15 from a normal distribution gives X = 10 and s2 = 25. Find 90% confidence interval for µ.

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