AHL 4.16—Confidence intervals Questionbank

Practice AHL 4.16—Confidence intervals with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A sample of 9 households records their monthly water usage (in liters): 200, 210, 220, 230, 240, 250, 260, 270, 280.

Calculate the sample mean, $\bar{x}$ , and the sample sample standard deviation, $s_{n-1}$ .

[2]

Find the $95 \%$ confidence interval for the population mean, $\mu$ , assuming the population standard deviation is unknown.

[3]

Sketch the $95 \%$ confidence interval on a number line.

[2]

Interpret the confidence interval in the context of the problem.

[1]

Question 2

HLPaper 1

A sample of 11 batteries has lifetimes (in hours): 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150.

Find the sample mean and sample variance, $s_{n-1}^{2}$ .

[3]

Construct a $95 \%$ confidence interval for the population mean, $\mu$ .

[3]

Determine the minimum sample size needed for a $95 \%$ confidence interval with a margin of error of 5 hours, using the sample standard deviation.

[3]

Question 3

HLPaper 2

A company measures the length (in cm) of a sample of 10 products: $30,32,33,34,35$ , $36,37,38,39,40$ . The population standard deviation is known to be $\sigma=3.5$ .

Find the $90 \%$ confidence interval for the population mean, $\mu$ .

[3]

Determine the sample size required to achieve a margin of error of 1 cm at a $90 \%$ confidence level.

[3]

If the sample standard deviation is used instead of $\sigma$ , explain how the confidence interval in part (a) would change.

[3]

Question 4

HLPaper 1

A coffee shop records the waiting time (in minutes) for a random sample of 30 customers, with $\sum t=135$ and $\sum t^{2}=650$ , where $t$ is the waiting time.

Find unbiased estimates for the mean and variance of the waiting time.

[3]

Assuming the waiting times are normally distributed, find the $95 \%$ confidence interval for the population mean waiting time, $\mu$ .

[3]

Sketch the $95 \%$ confidence interval on a number line.

[3]

State one assumption required for the confidence interval to be valid.

[1]

Question 5

HLPaper 2

Two years of temperature data (in ${ }^{\circ} \mathrm{C}$ ) for a city are recorded for 8 days: Year 1: 20, 22, 24, 25, 26, 28, 30, 32 Year 2: 22, 24, 26, 27, 29, 31, 33, 35

Conduct a paired $t$ -test at the $10 \%$ significance level to determine if the mean temperature differs between the two years. State the hypotheses, test statistic, and conclusion.

[5]

Find the $95 \%$ confidence interval for the mean difference in temperature.

[3]

Sketch the confidence interval for the mean difference on a number line.

[3]

Question 6

HLPaper 2

The continuous random variable X has probability density function $f$ given by

$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\text{otherwise}}} \end{array}} \right.$

Find ${\text{P}}\left( {X < 1} \right)$ .

[3]

Show that $a = \frac{2}{3}$ .

[3]

Given that ${\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)$ ,and that 0.25 < s < 0.4 , find the value of s.

[7]

Question 7

HLPaper 2

A researcher is studying the effect of a new drug on blood pressure. The population standard deviation of blood pressure reduction is known to be 8 mmHg. A sample of 64 patients is treated with the drug, and the mean reduction in blood pressure is found to be 15 mmHg.

Calculate a 99% confidence interval for the mean reduction in blood pressure.

[2]

The researcher is concerned that the blood pressure reductions may not be normally distributed. State, with a reason, whether the calculation of the confidence interval in part a is still valid.

[2]

Question 8

HLPaper 2

A researcher is studying the effect of a new fertilizer on plant growth. In a sample of 30 plants, the mean increase in height is 15 cm with a sample variance of 9 cm².

Find the unbiased estimate of the population variance.

[2]

Find a 99% confidence interval for the mean increase in height of the plants, stating any assumptions you make.

[3]

The researcher claims that the fertilizer increases plant height by more than 17 cm on average. Comment on this claim in the light of your confidence interval.

[2]

Question 9

HLPaper 1

A manager wishes to check the meanmass of flour put into bags in his factory. He randomlysamples 10 bags and finds the meanmass is 1.478 kg and the standard deviation of thesample is 0.0196 kg.

Find ${s_{n - 1}}$ for this sample.

[2]

Find a 95 % confidence interval for the population mean, giving your answer to4 significant figures.

[2]

The bags are labelled as being 1.5 kg mass. Comment on this claim with reference toyour answer in part (b).

[1]

Question 10

HLPaper 1

The continuous random variable X has a probability density function given by

$f(x) = \left\{ {\begin{array}{*{20}{l}} {k\sin \left( {\frac{{\pi x}}{6}} \right),}&{0 \leqslant x \leqslant \,6} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.$ .

Find the value of $k$ .

[4]

By considering the graph of f write down the mean of $X$ ;

[1]

By considering the graph of f write down the median of $X$ ;

[1]

By considering the graph of f write down the modeof $X$ .

[1]

Show that $P(0 \leqslant X \leqslant 2) = \frac{1}{4}$ .

[4]

Hence state the interquartile range of $X$ .

[2]

Calculate $P(X \leqslant 4|X \geqslant 3)$ .

[2]