IB Mathematics Applications & Interpretation Topic AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

A random variable $Z$ is normally distributed with mean $10$ and variance $4$ . Let $W = 3 Z_{1} + 2 Z_{2}$ , 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 $55$ .

[2]

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

[2]

Show that the estimator $\overset{μ}{^} = \frac{Z _{1} + Z _{2}}{2}$ is unbiased for the mean of $Z$ .

[2]

Question 2

HLPaper 1

A machine manufactures screws whose lengths are normally distributed with mean $4.5 mm$ and variance $0.09 mm^{2}$ . Treat the lengths in a sample as independent.

An inspector selects four screws.

Find the expected value of the combined length of the four screws.

[2]

Calculate the standard deviation for the combined length.

[2]

Find the probability that the combined length lies between $17.4 mm$ and $18.6 mm$ .

[2]

Sketch the distribution of the combined length, shading the interval from $17.4 mm$ to $18.6 mm$ .

[2]

Show that the sample mean of the screw lengths is an unbiased estimator of the population mean.

[2]

Question 3

HLPaper 1

The number of minutes required by students to finish a statistics quiz is normally distributed with a mean of 25 minutes and a standard deviation of 3 minutes. A teacher randomly chooses three students and notes the sum of their completion times.

Find the expected combined time for the three students to finish the quiz.

[2]

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

[2]

Find the probability that the combined time taken by the three students is greater than 80 minutes.

[3]

The teacher gives a reward if the combined time is under 70 minutes. Sketch the normal distribution of the combined time, shading the region $t < 70$ .

[3]

Question 4

HLPaper 2

A workshop manufactures cylindrical tins whose diameters are mutually independent and follow a normal distribution with mean $5.8 cm$ and variance $0.16 cm^{2}$ . A quality controller selects five tins and adds their diameters.

Determine the expected total diameter of the five tins.

[2]

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

[2]

The quality controller rejects consignments when the total diameter of five tins is below $28 cm$ . Find the probability that a consignment is rejected.

[3]

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

[3]

If the sample size increases to seven tins and the rejection threshold is changed to $39.2 cm$ (scaling $28 cm$ by $\frac{7}{5}$ ), find the probability of rejection.

[3]

Question 5

HLPaper 2

In a water-treatment plant, a probe outputs a raw voltage $R$ (in V). The controller converts this to a pH reading $P$ using the linear calibration model $P = m R + c$ , where $m$ and $c$ are constants.

Find expressions for $E (P)$ and $Var (P)$ in terms of $m$ , $c$ , $E (R)$ and $Var (R)$ . State the distribution of $P$ if $R$ is normally distributed.

[6]

The raw voltage is modelled as $R \sim N (2, 0. 5^{2})$ and the calibration uses $m = - 2$ and $c = 11$ .

Calculate $E (P)$ and the standard deviation of $P$ .

Using technology, find $P (P > 8)$ and interpret this probability in the context of the water-treatment plant.

[7]

IB Mathematics Applications & Interpretation AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators Questionbank

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IB Mathematics Applications & Interpretation AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 4.14—linear Transformation of a Single RV, E(X) and VAR(X), Unbiased Estimators Questionbank

Unlock all 30 questions with a free account