AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators Questionbank

Practice AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

A random variable $Z$ is normally distributed with mean 10 and variance 4 . A linear transformation is defined as $W=3 Z_{1}+2 Z_{2}$ , where $Z_{1}$ and $Z_{2}$ are independent observations of $Z$ .

Find the expected value of $W$ .

[2]

Calculate the variance of $W$ .

[2]

Find the probability that $W$ is between 45 and 55 .

[3]

Sketch the distribution of $W$ , shading the region where $45<W<55$ .

[3]

Show that the estimator $\hat{\mu}=\frac{Z_{1}+Z_{2}}{2}$ is unbiased for the mean of $Z$ .

[2]

Question 2

HLPaper 2

A quality control officer tests the resistance (in ohms) of resistors produced in a factory, which follows a normal distribution with mean 100 ohms and variance 16 ohms $^{2}$ . A sample of 12 resistors is taken, with $\sum x=1194$ ohms and $\sum x^{2}=119108$ ohms $^{2}$ .

Calculate an unbiased estimate for the population variance.

[3]

Construct a $95 \%$ confidence interval for the population mean resistance, using the sample standard deviation.

[3]

The factory claims the mean resistance is 98 ohms. Comment on the validity of this claim using the confidence interval.

[2]

If the total resistance of three resistors is measured, find the probability that it exceeds 305 ohms.

[3]

Question 3

HLPaper 2

Let $X$ be a random variable following a normal distribution with mean $\mu=12$ and variance $\sigma^{2}=9$ . A linear transformation is defined as $Y=2 X_{1}+3 X_{2}-X_{3}$ , where $X_{1}, X_{2}, X_{3}$ are independent observations of $X$ . A researcher collects a sample of 10 observations of $Y$ , with $\sum y=109.5$ and $\sum y^{2}=1236.25$ .

Find the expected value and variance of $Y$ .

[3]

Calculate an unbiased estimate for the population variance of $Y$ using the sample data.

[3]

Construct a $99 \%$ confidence interval for the population mean of $Y$ , using the sample variance.

[4]

Perform a chi-squared goodness-of-fit test at the $5 \%$ significance level to determine if the sample data for $Y$ follows a normal distribution with the mean and variance from part (a). The data is grouped into 5 intervals with observed frequencies: 1,3 , $4,2,0$ . Expected frequencies under the null hypothesis are $0.8,2.7,4.0,2.3,0.2$ . State the hypotheses, degrees of freedom, chi-squared statistic, and conclusion. [6

[6]

Sketch the distribution of $Y$ , shading the region corresponding to $Y>30$ .

[3]

Question 4

HLPaper 1

Let $W_{1} \sim$ Poisson(5) and $W_{2} \sim$ Poisson(7) be independent random variables. Define $Z=3 W_{1}-2 W_{2}+4$ .

Calculate the expected value and variance of $Z$ .

[3]

Using a normal approximation, find the probability that $Z \leq 0$ .

[3]

A sample of 8 observations of $Z$ yields $\sum z=72$ and $\sum z^{2}=680$ . Find an unbiased estimate for the population mean of $Z$ .

[2]

Show that the sample mean $\bar{Z}$ is an unbiased estimator for the population mean of $Z$ .

[2]

Perform a hypothesis test at the $1 \%$ significance level to test if the population mean of $Z$ equals 5, using the sample mean from part (c) and assuming the population variance is as calculated in part (a). State the hypotheses, test statistic, p-value, and conclusion.

[6]

Question 5

HLPaper 2

The continuous random variable X has probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\text{otherwise}}} \end{array}} \right.$

Find ${\text{P}}\left( {X < 1} \right)$ .

[3]

Show that $a = \frac{2}{3}$ .

[3]

Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ ,and that 0.25 < s < 0.4 , find the value of s.

[7]

Question 6

HLPaper 3

Two independent random variables $X$ and $Y$ follow Poisson distributions.

Given that ${\text{E}}\left( X \right) = 3$ and ${\text{E}}\left(Y \right) = 4$ , calculate

${\text{E}}\left( {2X + 7Y} \right)$ .

[2]

Var $\left( {4X - 3Y} \right)$ .

[3]

${\text{E}}\left( {{X^2} - {Y^2}} \right)$ .

[4]

Question 7

HLPaper 2

A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$ , such that ${\text{P}}(X < 30.31) = 0.1180$ and ${\text{P}}(X > 42.52) = 0.3060$ .

Find $\mu$ and $\sigma$ .

[6]

Find ${\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)$ .

[2]

Question 8

HLPaper 2

A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let ${\text{P}}(X = n)$ be the probability that Kati obtains her third voucher on the $n{\text{th}}$ bar opened.

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that ${\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}$ for $n \geqslant 3,{\text{ }}n \in \mathbb{N}$ .

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.

Show that ${\text{P}}(X = 3) = 0.001$ and ${\text{P}}(X = 4) = 0.0027$ .

[3]

Find the values of the constants $a$ and $b$ .

[5]

Deduce that $\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}$ for $n > 3$ .

[4]

(i) Hence show that $X$ has two modes ${m_1}$ and ${m_2}$ .

(ii) State the values of ${m_1}$ and ${m_2}$ .

[5]

Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.

[3]

Question 9

HLPaper 3

Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.

${\text{7.7}}\quad {\text{7.5}}\quad {\text{8.4}}\quad {\text{8.8}}\quad {\text{7.3}}\quad {\text{9.0}}\quad {\text{7.8}}\quad {\text{7.6}}$

Assume that these weights form a random sample from a $N(\mu ,{\text{ }}{\sigma ^2})$ distribution.

Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis ${{\text{H}}_0}:\mu = 7.5$ .

Determine unbiased estimates for $\mu$ and ${\sigma ^2}$ .

Use a two-tailed test to determine the $p$ -value for the above results.

Interpret your $p$ -value at the 5% level of significance, justifying your conclusion.

Question 10

HLPaper 2

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.

It is found that 5% of the male runners complete the marathon in less than ${T_1}$ minutes.

The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.

Find the probability that a runner selected at random will complete the marathon in less than 3 hours.

[2]

Calculate ${T_1}$ .

[2]

Find the standard deviation of the times taken by female runners.

[4]

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that ${\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}$ for $n \geqslant 3,{\text{ }}n \in \mathbb{N}$ .

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.