4.16 Confidence intervals questions for IB Maths AI

State why a confidence interval based on the normal distribution, rather than the $t$-distribution, is appropriate here.

Find a 95% confidence interval for the mean mass of flour in a bag. Give the endpoints correct to one decimal place.

The manager believes that the machine is still set correctly. Use your answer to part (b) to comment on her belief.

Find unbiased estimates of the mean and the standard deviation of the mass of the contents of a box.

Find a 95% confidence interval for the mean mass of the contents of a box. Give the endpoints correct to four significant figures.

Use your interval to comment on the label.

The consumer group decides to carry out a hypothesis test, at the 1% significance level, to determine whether the mean mass of the contents is less than 500 g.

(i) Write down the null and alternative hypotheses.
(ii) Find the $p$-value for this test.
(iii) State the conclusion of the test. Give a reason for your answer.

State one assumption, other than normality, needed for your answers to parts (b) and (d) to be valid.

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.

Find unbiased estimates of the mean and the variance of the heights of the seedlings.

Find a 90% confidence interval for the mean height of the seedlings.

Explain why Priya should use the $t$-distribution to find this interval.

Explain why a confidence interval based on the normal distribution can be used even though the distribution of delivery times is not known.

Find
(i) a 95% confidence interval;
(ii) a 99% confidence interval
for the mean delivery time.

Explain why the interval in part (b)(ii) is wider than the interval in part (b)(i).

Valentina says that Fratelli Pizza's advertisement is false. Comment on her statement.

Assume that the mean delivery time is 30 minutes. Find the probability that the mean of a random sample of 36 delivery times is at least 31.8 minutes.

Fratelli Pizza wants to estimate its mean delivery time with a 95% confidence interval of width at most 3 minutes. Find the least number of deliveries it must sample.

Find $s_{n-1}$, the unbiased estimate of the population standard deviation.

Find a 99% confidence interval for the mean battery life. Give the endpoints correct to four significant figures.

Comment on the manufacturer's claim.