Exercises for Question Type 3: Estimating the mean and variance using unbiased estimators

Question 1

Skill question

The following frequency distribution table represents a data set:

Value ( $x$ )	4	5	6	7	9
Frequency ( $f$ )	8	12	6	8	2

Using this data set, calculate the unbiased estimator for the population variance.

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Question 2

Skill question

Assume a distribution has parameters $a$ and $b$ satisfying $E (X) = a + b$ and $Var (X) = a$ . Using the sample mean $\overset{x}{ˉ} = 5.6111$ and sample variance $s^{2} = 1.8418$ from the data, find the method-of-moments estimators for $a$ and $b$ .

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Question 3

Skill question

A statistical study calculated the sample variance of a set of data to be $s^{2} = 1.8418$ .

Calculate the sample standard deviation $s$ . Give your answer correct to four decimal places.

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Question 4

Skill question

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a normal distribution with mean $μ$ and variance $a$ .

Consider the estimator for $b = μ - a$ defined by $\hat{b} = \overset{ˉ}{X} - \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ .

It is given that $\hat{b}$ is an unbiased estimator for $b$ and that its variance is $Var (\hat{b}) = \frac{a}{n} + \frac{2 a ^{2}}{n - 1}$ .

Show that the estimator $\hat{b}$ is consistent for $b$ .

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Question 5

Skill question

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population with mean $a + b$ . Let $\overset{ˉ}{X}$ be the sample mean and let $\overset{a}{^}$ be an unbiased estimator for $a$ .

Consider the estimator $\hat{b} = \overset{ˉ}{X} - \overset{a}{^}$ . Determine whether $\hat{b}$ is an unbiased estimator for $b$ .

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Question 6

Skill question

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a distribution such that $E (X_{i}) = a + b$ and $Var (X_{i}) = a$ , where $a, b \in R$ and $a > 0$ .

Determine whether the estimator $\tilde{b} = \overset{ˉ}{X} - \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ is unbiased for $b$ . If it is biased, find its bias.

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Question 7

Skill question

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size $n$ from a normal distribution with mean $μ$ and variance $a$ . Let $\overset{ˉ}{X}$ be the sample mean and $S^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ be the unbiased sample variance. It is given that $\overset{ˉ}{X}$ and $S^{2}$ are independent.

Given $c = \frac{1}{n - 1}$ , derive the variance of the unbiased estimator $\hat{b} = \overset{ˉ}{X} - c i = 1 \sum n (X_{i} - \overset{ˉ}{X})^{2}$ in terms of $a$ and $n$ .

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Question 8

Skill question

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size $n > 1$ from a distribution with mean $E (X_{i}) = a + b$ and variance $Var (X_{i}) = a$ , where $a$ and $b$ are constants such that $a > 0$ .

Find the constant $c$ such that the estimator $\hat{b}_{c} = \overset{ˉ}{X} - c i = 1 \sum n (X_{i} - \overset{ˉ}{X})^{2}$ is an unbiased estimator for $b$ .

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Question 9

Skill question

Given the following data values with frequencies: 4 (8 times), 5 (12 times), 6 (6 times), 7 (8 times), and 9 (2 times), compute the unbiased estimator for the population mean.

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Question 10

Skill question

Let $X_{1}, \dots, X_{n}$ be iid from a distribution with $E (X) = a + b$ and $Var (X) = a$ . Show that the estimator $\overset{a}{^} = \frac{1}{n - 1} i = 1 \sum n (X_{i} - \overset{ˉ}{X})^{2}$ is unbiased for $a$ .

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Question 11

Skill question

A sample of size $n = 36$ is taken from a population. For this data set, the sum of squared deviations from the mean is $\sum_{i = 1}^{36} (x_{i} - \overset{x}{ˉ})^{2} = 64.4640$ and the unbiased estimator of the population variance is $s^{2} = 1.8418$ .

Compute the biased estimator of the variance (using denominator $n$ ) and find the numerical bias by comparing it with the unbiased estimator.

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IB Mathematics Applications & Interpretation Question Type 3: Estimating the Mean and Variance Using Unbiased Estimators Exercises