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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A machine manufactures screws whose lengths are normally distributed with mean and variance . Treat the lengths in a sample as independent.
An inspector selects four screws.
(a) Find the expected value of the combined length of the four screws.
(b) the standard deviation for the combined length.
(e) If the sample size increases to seven tins and the rejection threshold is changed to (scaling by ), find the probability of rejection.
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.
Assume the students' completion times are independent.
(a) Find the expected combined time for the three students to finish the quiz.
(b) the standard deviation of the combined time taken by the three students.
(c) Find the probability that the combined time taken by the three students is greater than 80 minutes.
A workshop manufactures cylindrical tins whose diameters are mutually independent and follow a normal distribution with mean and variance . A quality controller selects five tins and adds their diameters.
(a) the expected total diameter of the five tins.
(b) Find the variance of the total diameter of the five tins.
(c) The quality controller rejects consignments when the total diameter of five tins is below . Find the probability that a consignment is rejected.
(d) the distribution of the total diameter, showing the rejection region.
In a water-treatment plant, a probe outputs a raw voltage (in V). The controller converts this to a pH reading using the linear calibration model , where and are constants.
(a) Find expressions for and in terms of , , and . the distribution of if is normally distributed.
(b) The raw voltage is modelled as and the calibration uses and .
and the standard deviation of .
Using technology, find and this probability in the context of the water-treatment plant.
Dispensing volumes for a café syrup pump are assumed to be normally distributed. A random sample of 10 drinks gives a mean syrup volume of 18.7 ml and a standard deviation of 1.5 ml calculated using divisor , so .
(b) a 95 % confidence interval for the true mean syrup volume, presenting your result to 4 significant figures.
(c) The manager claims that the mean syrup volume is 20 ml. this claim using your confidence interval.
(d) the distribution of the combined length, shading the interval from to .
(e) whether the sample mean of the screw lengths is an unbiased estimator of the population mean. what this means in the context of the screw lengths.
(d) The teacher gives a reward if the combined time is under 70 minutes. the normal distribution of the combined time, shading the region .
(a) the unbiased estimate of the population standard deviation, , for these data.
