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Problem Set 1
Statistical Inference: Estimation
V. Troster and A.Virbickait˙e
- (Exam of September 2013) For a population with mean μ and variance σ^2 an estimator of μ is proposed that is calculated from a random sample of 4 independent observations:
μˆ =
y 1 +
y 2 +
y 3 + ay 4
a) Calculate the value of a such that the estimator ˆμ is unbiased.
b) Calculate the variance of ˆμ.
- Consider the following estimator: ˆμ = 12 yi + 12 yj , where yi and yj are two observations randomly selected from a random sample of n observations (n ≥ 2). It is known that E [ˆμ] = μ and Var [ˆμ] = σ^2 /2.
a) Is ˆμ an unbiased estimator? Justify your answer.
b) Is ˆμ a consistent estimator? Justify your answer.
c) Are all unbiased estimators also consistent?
- Let y 1 and y 2 be a random sample from a normally distributed population N (μ, σ^2 ). Consider the following estimators of μ:
y¯ =
y 1 +
y 2
μ ˆ 1 =
y 1 +
y 2
μ ˆ 2 =
y 1 +
y 2.
a) Show that all three estimators are unbiased.
b) Which of the estimators is the most efficient?
c) Which of the three estimators would you choose to estimate μ and why?
- The table below contains various confidence intervals for the population mean for a certain random variable. These confidence intervals have been calculated using different confidence levels and two different sample sizes (10 and 100 observations). Comment on the differences in the intervals with respect to the confidence level, sample size and the width of the interval.
n = 10 n = 100 Confidence level Lower limit Upper Limit Lower Limit Upper Limit 90% 7.41 12.59 9.26 10. 95% 6.80 13.20 9.11 10. 99% 5.40 14.60 8.83 11.
- For a normally distributed random variable N (5, 1) we have a random sample of n = 25 observations. Calculate the following probabilities:
a) Pr(¯y > 5)
b) Pr(¯y < 4 .7)
c) Pr(4. 9 < y <¯ 5 .1)
- Using the information from the previous exercise, answer the following questions:
a) What should be the value of μ such that Pr(¯y > 10) = 0.5?
b) What would be the value of Pr(¯y > 5 .1) if instead of a sample size n = 25 we had a sample of n = 1000 observations?
- (Exam of February 2017) For a population y ∼ N (60, σ^2 = 20), a random sample was obtained with sample size of n = 25, {y 1 , y 2 ,... , y 25 }. Answer the following questions:
a) Let ¯y be the sample mean of y, calculate Pr (58 < ¯y < 61 .5).
b) Let μ be the population mean of y, what is the value of μ such that Pr (¯y < 58) = 0.05?
- (Exam of January 2018) The CEO of a company thinks that 20% of all orders are placed by the new buyers. In order to calculate the proportion of orders placed by the new buyers, a random sample of 85 orders is analyzed. Suppose that the population proportion of the orders placed by the new buyers is known and is p = 0.2. What is the probability that the sample proportion (¯p) of orders placed by the new buyers is less than 0.15 in this sample?
- For a population y ∼ N (μ, σ^2 ), a random sample was obtained with sample size of n, {y 1 , y 2 ,... , yn}. We want to estimate the population variance σ^2. We know that
∑^ n
i=
(yi − ¯y)^2 σ^2
∼ χ^2 n− 1 ,
a) Can we say that the population mean rent is equal to 680e?
b) Can we say that the population mean rent is equal to or more than 800e?
c) Considering the people who live in rental apartments in that city, is it possible that the unemployment rate is larger than the overall country’s unemployment rate of 20%?
- (Exam of September 2014) In order to study gas consumption by cars in a certain city, a random sample of 20 drivers was selected. They were asked which type of gas they are using and how much they spend on gas per week. Figure 1 contains the responses of the 20 drivers. Next, Figure 2 contains the descriptive statistics obtained via Excel for the spending variable.
Knowing that the weekly spending on gas by the people in that city is normally distributed N (μ, 400), produce a point and interval estimator of the average weekly spending on gas using the 95% confidence level.
- (Exam of February 2015) Using a random sample of n = 10 observations, obtained from a population with mean μ and variance σ^2 , the following estimators for μ are defined:
μˆ 1 =
(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 )
μˆ 2 =
(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 )
μˆ 3 = x 1 + x 2 + x 3 + x 4 + x 5 − x 6 − x 7 − x 8 − x 9.
a) Determine if the proposed estimators are unbiased.
b) Which estimator is the most efficient?
c) If n = 10, ¯x = 18, ˆs^2 = 9, calculate the confidence interval for μ using the 95% confidence level.
- In a random sample of 100 toys, produced by a company, 20% do not meet the required quality standards. Construct a confidence interval using the 95% confidence level for the population proportion of toys that do not meet the required quality standards.
- Car rental agency wants to estimate how many kilometers on average their cars drive every day. For that reason a random sample of 71 cars is selected, where the sample mean of 165 km/day and standard deviation of 45 km/day. Given that the number of kilometers per day is normally distributed, calculate a confidence interval for the mean using the 95% confidence level.
- (Exam of February 2018) Let θˆ 1 be an estimator of the parameter θ, where θˆ 1 satisfies E
θˆ 1
= 34 θ. If for a random sample it was obtained that θˆ 1 = 9000, calculate based on θˆ 1 an estimate of θ that is unbiased. Justify your answer.
Solutions
- a) 1/12; b) (42/144)σ^2.
- a) Yes, since E(ˆμ) = μ; b) No, since limn→∞ ECM (ˆμ) = σ^2 / 2 6 = 0; c) No, since we just saw a counter-example of an unbiased estimator that is inconsistent.
- b) ¯y; c) ¯y because it is the most efficient one.
- ⇑ (1 − α) ⇒⇓ Margin of Error of the CI, given n; ⇑ n ⇒⇓ Margin of Error of the CI, given (1 − α).
- a) 0.5; b) 0.0668; c) 0.3830.
- a) μ = 10; b) 0.
- a) 0.941; b) μ = 59.4713.
- 0.1251.
