4.18 Further hypothesis testing questions for IB Maths AI

Write down the null and alternative hypotheses.

Find the $p$-value for this test.

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

Write down the critical region for this test.

State two reasons why a Poisson distribution might be a suitable model for the number of mushrooms in a plot.

Find the probability that a plot contains exactly 4 mushrooms.

Find the probability that a plot contains fewer than 3 mushrooms.

Ingrid inspects 4 plots in different parts of the forest. Find the probability that every one of these plots contains at least one mushroom.

After a period of heavy rain, Ingrid believes that the mean number of mushrooms per square metre has increased. She will count the mushrooms in one 8 m² plot and test her belief at the 5% significance level.

Write down the null and alternative hypotheses for this test.

Find the critical region for this test.

After the rain, the mean number of mushrooms is actually 0.8 per square metre. Find the probability that Ingrid's test makes a Type II error.

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.

Write down the null and alternative hypotheses for Siobhan's test.

Find the $p$-value for this test.

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

Find the probability that no counts are recorded in a 5-second interval.

Find the length of time, in seconds, for which the probability of recording at least one count is 0.9.

A sample of rock is placed next to the detector. The rock emits radiation that adds an independent 1.1 counts per second, on average, to the background count.

Find the probability that more than 20 counts are recorded in a 10-second interval.

Teresa uses the detector to decide whether a sample is radioactive. She records the number of counts, $N$, in 10 seconds with the sample in place. Her hypotheses are

${\mathrm{H}}_0$: the sample is not radioactive (the mean count in 10 seconds is 4)
${\mathrm{H}}_1$: the sample is radioactive (the mean count in 10 seconds is greater than 4).

She rejects ${\mathrm{H}}_0$ if $N\ge 9$.

(i) Find the probability that Teresa makes a Type I error.
(ii) Find the probability that Teresa makes a Type II error when she tests the rock in part (c).

Suggest how Teresa could reduce the probabilities of both types of error.

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.