Ejercicios Estadísticos: Estimación de Parámetros, Apuntes de Estadística

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Grau d'Administració i Direcció d'Empreses Grau d'Economia

Statistics II

Set of exercises 2

Unit 2 - Estimation

Professors: Mikel Esnaola David Moriña

1. Determine, in each case, if the values shown are parameters or point estimations. Explain why: (a) The INE, according to the results of the Encuesta sobre la Población Activa for the rst quarter of 2009, claims that the unemployment rate in Catalunya is of 16. 16 %. (b) A telemarketing company uses a machine that dials randomly chosen phone numbers from a given town. Only 47 out of the rst 100 numbers dialed are listed in the phone book. As a matter of fact, this is not surprising as only a 52% of all the phone numbers in town are listed in the phone book.
2. We want to study the mean parameter, μ. Compute the bias for each of the estimators below:

a) μˆ =

x 1 + x 2 + x 3 + x 4 4

b) μˆ =

x 1 − x 2 + x 3 − x 4 4 c) μˆ =

x 1 + x 2 + x 3 + x 4 + x 5 + x 6 3

d) μˆ =

x 1 − x 2 + x 3 + x 4 − x 5 − x 6 4 e) μˆ = x 1 − x 2 + x 3 3

f) μˆ = x 1 + x 2 + x 3 3 g) μˆ =

x 1 + x 2 + · · · + xn n − 1

h) μˆ =

x 1 + x 2 + · · · + xn n + 1 i) μˆ =

x 1 − x 2 + x 3 − x 4 + · · · + xn n j) μˆ =

x 1 + 2x 2 + 3x 3 + · · · + nxn n(n + 1)

3. Let X be a random variable with density function given by:

f (x) = θe−θx, x ≥ 0 , θ > 0

Compute the bias of the following estimators of θ:

a) ˆθ =

2 x 1 + 4x 2 3

b) θˆ =

4 x 1 + 5x 2 3 c) θˆ =

x 1 + 3x 2 4

d) θˆ =

x 1 + x 2 + · · · + xn n Hint: E[X] = (^1) θ

4. Let X ∼ N (μ, σ^2 ). Compute the Mean Quadratic Error, MQE(μˆ), in each of the pairs of estimators below and determine, in each case, which one of the two estimators is more ecient:

a) μˆ 1 =

x 1 + 2x 2 + x 3 2

b) μˆ 2 =

x 1 + x 2 − x 3 + x 4 + x 5 + x 6 2 c) ˆμ 1 =

x 1 + x 2 + x 3 3

d) μˆ 2 =

x 1 − x 2 + x 3 + x 4 2

5. We want to study the mean parameter, μ. Which ones of the estimators below are asymp- totically unbiased?

a) μˆ = x 1 + x 2 − x 3 n

b) μˆ = x 1 + nxn n c) μˆ =

x 1 + x 2 + · · · + xn n − 1

d) μˆ =

x 1 + x 2 + · · · + xn n + 1

Assuming that thickness is a Normal random variable, what population parameter is infor- mative about the uctuation? What estimator can be proposed for this parameter?

13. A manufacturer claims that the percentage of faulty items in any lot of the articles he produces is less than 1%. The quality control department draws a sample of size 350 from the lot and nds 3 faulty pieces. What is the estimate of the proportion of faulty items?

14. We want to study the volatility in the stock market of the returns of the stocks of a given company. To do so, we have the following series consisting of the returns of two weeks

15. 01 , − 0. 02 , − 0. 08 , 0. 12 , 0. 04 , − 0. 1 , 0. 06 , 0. 01 , − 0. 01 , − 0. 01

What estimate can be produced to approximate the volatility?

15. Two candidates (A, B) run for election. One of the candidates, A, decides to conduct a survey. One hundred and fty people are randomly chosen and 90 say that will vote for him. Construct a 95% condence interval. From this information, can he be condent in winning the election?

16. A manufacturer claims that the percentage of faulty items in any lot of the articles he produces is 1%. A random sample of 200 articles is selected and 8 are found to be faulty. Find 95% and 99% condence intervals for the true proportion of faulty items. Based on these results, what can you say about the manufacturer's claim?

17. A physician is interested in the proportion of men that smoke and develop lung cancer. The physician wants to select a sample of smokers and observe whether they develop cancer or not. What has to be the sample size so that with a 95 % probability the dierence between the sample proportion and the true proportion is less than 0.02?

18. Auditing rms usually select a random sample from the accounts of a bank and verify the balance sheets reported by the bank. One of such rms wants to estimate the proportion of accounts for which the real gures and the information provided by the bank do not coincide. How many accounts must be chosen at least so that, with a condence of 95%, the dierence between the sample proportion and the true proportion is not more than 0.02?

19. In a random sample with 100 individuals the average level of glucose in blood is 110 mg/cc. We know that the standard deviation is 20 mg/cc. Construct a 90% the level of glucose in blood of the population. What is the maximum error in this estimation?

20. A random sample containing the hourly wages of nine mechanics gives:

21. 5 , 11 , 9. 5 , 12 , 10 , 11. 5 , 13 , 9 , 8. 5

Assuming that the sample has been drawn from a Normal population, construct condence intervals for the mean wage with α = 0.05 and α = 0. (a) if the population variance is known: σ^2 = 1.5. (b) if the population variance is unknown σ^2 =?.

21. Scientists in a purifying plant analyze the contents of lead in water. After studying 40 samples they obtain X = 3 and S^2 = 2. Construct a 95% condence interval for the average content of lead in water.
22. The level of cholesterol is a high risk factor in the development of heart diseases. One study is conducted to determine the level of cholesterol in blood. A sample of 96 patients is randomly

drawn and the results obtained show an average of 170.81 mg/dl with a standard deviation of 30.55 mg/dl. Construct a 95% condence interval for the average level of cholesterol in blood.

23. Given a Normal random variable with a standard deviation of 4, we want to construct a 95% condence interval for the average. What has to be the size of the sample so that the error is less than 0.5?

24. The thickness of the metal pieces that a machine produces is expected to present some uctu- ation. A random sample of 12 pieces is selected and the thickness of each of them is recorded, which yields

25. 6 , 11. 9 , 12. 3 , 12. 8 , 11. 8 , 11. 7 , 12. 4 , 12. 1 , 12. 3 , 12. 0 , 12. 5 , 12. 9

Assuming that thickness is a Normal random variable, obtain a 95% condence interval for the variance of thickness.

25. In each case, obtain estimators of the unknown parameters using the method of moments. Once the estimator is found, use it to produce a point estimate of the parameter using the sample data in parenthesis: (a) X ∼ P oiss(λ) where λ is the unknown parameter. ({2, 4, 7, 3, 5}) (b) X ∼ U nif [0, θ] where θ is the unknown parameter. ({1.2, 7, 3.3, 5, 8.5}) (c) X random variable with density function given by:

f (x) = λe−λx, x ≥ 0

where λ is the unknown parameter. ({1.4, 2, 4.6, 3})

26. In each case, obtain estimators of the unknown parameters using the maximum likelihood method. Once the estimator is found, use it to produce a point estimate of the parameter using the sample data in parenthesis: (a) X random variable with density function given by:

f (x) = λe−λx, x ≥ 0 , λ > 0

where λ is the unknown parameter. ({2, 5, 4.4, 3, 1.9, 2}) (b) X random variable with density function given by:

f (x) = θxθ−^1 , 0 < x < 1 , θ > 0

where n is the unknown parameter. ({0.9, 0.6, 0.8, 0.75, 0.3, 0.8, 0.8}) (c) X random variable with density function given by:

f (x) = θ(1 − x)θ−^1 , 0 < x < 1 , θ > 0

where θ is the unknown parameter. ({0.9, 0.6, 0.8, 0.75, 0.3, 0.8, 0.8}) (d) X ∼ Ber(p) where p is the unknown parameter. ({1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0}) (e) X ∼ Geom(p) where p is the unknown parameter. ({2, 6, 5, 8, 4 , 5, 6, 3, 8, 7})