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

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 1

The time, in minutes, taken by students to complete a mathematics quiz follows a normal distribution with a mean of 28 minutes and a standard deviation of 4 minutes. A teacher selects three students at random and records the total time taken by them to complete the quiz.

Find the expected total time for the three students to complete the quiz.

[2]

Calculate the standard deviation of the total time taken by the three students.

[2]

Find the probability that the total time taken by the three students exceeds minutes.

[3]

The teacher awards a prize if the total time is less than 80 minutes. Sketch the normal distribution of the total time, shading the region corresponding to this prize condition.

[3]

Question 6

HLPaper 2

A random variable $T$ is normally distributed with mean 100 and variance 64. A sample of 16 observations of $T$ is taken, with $\sum t=1612$ and $\sum t^{2}=162704$ . A linear transformation is defined as $S=2 T_{1}-T_{2}+3$ , where $T_{1}, T_{2}$ are independent observations of $T$ .

Find an unbiased estimate for the population variance of $T$ .

[3]

Construct a $98 \%$ confidence interval for the population mean of $T$ .

[4]

Perform a chi-squared test at the $10 \%$ significance level to test if the sample data for $T$ follows a normal distribution with mean 100 and variance 64. The data is grouped into 6 intervals with observed frequencies: $2,3,5,4,1,1$ , and expected frequencies: 1.8, 3.2, 5.0, 4.2, 1.5, 0.3. State the hypotheses, degrees of freedom, chi-squared statistic, and conclusion.

[6]

Find the probability that $S>105$ .

[3]

Sketch the distribution of $S$ , shading the region where $S>105$ .

[3]

Question 7

HLPaper 1

Let $Y$ be a Poisson random variable with mean 8 . A researcher observes the sum of five independent observations of $Y$ , denoted $T=Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}$ .

Find the expected value and variance of $T$ .

[2]

Using a normal approximation, calculate the probability that $T$ is at least

[3]

Show that the sample mean $\bar{Y}=\frac{T}{5}$ is an unbiased estimator for the population mean of $Y$ .

[2]

Sketch the normal approximation of $T$ , shading the region where $T \geq 45$ .

[3]

Question 8

HLPaper 1

Let $P \sim$ Poisson(6) and $Q \sim$ Poisson(4) be independent random variables. Define $R=2 P_{1}+P_{2}-Q_{1}$ , where $P_{1}, P_{2}, Q_{1}$ are independent observations of $P$ and $Q$ .

Calculate $E(R)$ and $\operatorname{Var}(R)$ .

[3]

Using a normal approximation, find the probability that $R>15$ .

[3]

A sample of 20 observations of $R$ has a mean of 8.2. Perform a hypothesis test at the $5 \%$ significance level to test if the population mean of $R$ equals 8 , assuming the population variance from part (a). State the hypotheses, test statistic, p-value, and conclusion.

[6]

Show that the estimator $\hat{\mu}=\frac{P_{1}+Q_{1}}{2}$ is unbiased for the mean of the combined population mean of $P$ and $Q$ .

[3]

Find the probability that $R_{1}+R_{2}>30$ , where $R_{1}, R_{2}$ are independent observations of $R$ .

[4]

Question 9

HLPaper 2

A biologist measures the length of leaves on a plant species, which follows a normal distribution with mean 15 cm and standard deviation 2 cm . A sample of 10 leaves is collected.

Given

\sum x=148.5

cm and

\sum x^{2}=2210.25

, find the unbiased estimate for the variance of the leaf lengths.

[3]

Construct a $95 \%$ confidence interval for the population mean length.

[3]

The biologist claims the population mean is 14.5 cm . Comment on the claim's validity using the confidence interval.

[2]

Find the probability that the combined length of two leaves exceeds 31 cm .

[3]

Question 10

HLPaper 2

A factory produces cylindrical cans with diameters following a normal distribution with mean 6.2 cm and variance 0.36 cm . The factory manager samples five cans and calculates their total diameter.

Determine the expected total diameter of the five cans.

[2]

Find the variance of the total diameter of the five cans.

[2]

The manager rejects batches if the total diameter of five cans is less than 30 cm . Find the probability of a batch being rejected.

[3]

Sketch the distribution of the total diameter, indicating the rejection region.

[3]

If the sample size increases to seven cans, find the probability of rejection, assuming the rejection threshold adjusts to 42 cm .

[3]

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