4.14 Linear transformations of random variables questions for IB Maths AI

Twelve adults enter the lift. Let $T$ be their total mass, in kg. Find the mean and the variance of $T$.

Find the probability that the total mass of the twelve adults exceeds the maximum load.

The building manager wants the probability that the total mass of the adults in the lift exceeds 1000 kg to be less than 1%. Find the greatest number of adults that should be allowed in the lift at one time.

Find an expression for $(s\circ p)(r)$.

Find the scaled score of Chiara, whose raw mark is 52.

Find an expression for $(s\circ p{)}^{-1}(x)$.

A grade 7 is awarded for a scaled score of 75 or more. Hence find the minimum raw mark needed for a grade 7.

Find $(s\circ p)(80)$ and comment on your answer.

The raw marks of the candidates are modelled by a normal distribution with mean 48 and standard deviation 9. Find the probability that a randomly chosen candidate is awarded a grade 7.

Find the mean and the standard deviation of the scaled scores.

Ingrid wants to model the number of boats arriving in one hour by a Poisson distribution.

State two conditions that must hold for this model to be appropriate.

Calculate the mean and the variance of the number of boats arriving in one hour, according to Ingrid's data.

Explain how your answers to part (b)(i) support the use of a Poisson model.

Ingrid models the number of boats arriving in one hour by a Poisson distribution with mean 3.2.

Find the probability that exactly 5 boats arrive in a one-hour period.

Given that at least 3 boats arrive in a two-hour period, find the probability that fewer than 6 boats arrive in that period.

The lock operates in 30-minute cycles and can take at most 4 boats in each cycle. Any extra boats must wait for the next cycle.

Find the probability that more than 4 boats arrive during one 30-minute cycle.

On one day the lock operates 20 cycles. The numbers of boats arriving in different cycles are independent.

Find the probability that more than 4 boats arrive in at least two of the cycles on that day.

The following summer a new marina opens upstream. Ingrid believes that the mean number of boats arriving per hour has increased. She will count the number of boats, $T$, arriving during one randomly chosen five-hour period and test her belief at the 5 % significance level.

State the null and alternative hypotheses for this test.

Find the critical region for this test.

During the five-hour period, 25 boats arrive at the lock.

State the conclusion of the test, in context. Give a reason for your answer.

Find the probability of a Type I error for this test.

In fact, after the marina opens, the mean number of boats arriving is 4.4 per hour.

Find the probability of a Type II error for this test.

Suggest how Ingrid could reduce the probability of a Type II error without increasing the significance level of the test.

Find $\mathrm{E}(B-T)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(B-T)$.

On a randomly chosen day, find the probability that a bus journey would be quicker than a tram journey.

In one week Amara travels by tram on each of her 5 working days. Find the probability that her total travel time for these five journeys is more than 160 minutes.

Use the nearest neighbour algorithm, starting at $A$, to find an upper bound for her total walking time.

By deleting vertex $A$ and using Prim's algorithm starting at $B$, find a lower bound for her total walking time.

Deleting vertex $D$ gives a lower bound of 42 minutes. Let $W$ be her least possible total walking time. Write down the best inequality for $W$.

The times, in minutes, that Siobhan spends inside the glasshouses are independent and normally distributed as follows.

Find the mean and the variance of $S$, the total time she spends inside the glasshouses.

Siobhan follows the route found in part (a). Find the probability that her whole visit, walking and time in the glasshouses, takes no more than 115 minutes.

The garden closes at 17:00. Find the latest time, to the nearest minute, that Siobhan can arrive at the entrance so that the probability that she finishes her visit before closing is at least 0.95.

Aroha will measure a stratified sample of 60 trout, stratified by region.

Find the number of trout she should measure from each region.

Let $\bar{L}$ be the mean length, in cm, of a random sample of 60 trout from the lake, and let $\mu$ be the mean length of all the trout in the lake.

Explain why $\bar{L}$ can be modelled by a normal distribution, and state this distribution in terms of $\mu$.

Find the probability that $\bar{L}$ is within 1.5 cm of $\mu$.

Find the smallest sample size for which the probability that the sample mean is within 1.5 cm of $\mu$ is at least 0.98.

Aroha measures a random sample of 60 trout. Their mean length is 34.6 cm.

Find a 95 % confidence interval for $\mu$.

Ten years earlier, the mean length of trout in Lake Moana was 32.5 cm.

Comment on whether Aroha's interval suggests that the mean length has changed.

Aroha also studies trout in Lake Kōwhai, where the standard deviation of lengths is unknown and the lengths are normally distributed. She measures the lengths, in cm, of a random sample of 10 trout.

Find unbiased estimates of the mean and the variance of the lengths of trout in Lake Kōwhai.

Find a 90 % confidence interval for the mean length of trout in Lake Kōwhai.

Explain why a $t$-distribution, and not a normal distribution, is used for this interval.

Trout in Lake Kōwhai can only be fished commercially if their mean length is greater than 30 cm.

Use Aroha's sample to test, at the 5 % significance level, whether the mean length of trout in Lake Kōwhai is greater than 30 cm.

The price, $V$ euros (EUR), paid for a trout from Lake Moana of length $L$ cm is given by $V=0.9L-6$. Assume that $L$ has mean 34.6 cm and standard deviation 6.2 cm.

Find the mean and the standard deviation of $V$.

A box contains 25 trout from Lake Moana, whose lengths are independent.

Find the mean and the standard deviation of the total price paid for the box.