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AHL 4.16—Confidence intervals - IB Questionbank | RevisionDojo

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 1

A store records daily sales (in dollars) over 25 days: $\sum x=6250, \sum x^{2}=1587500$ .

Find an unbiased estimate for the population mean, $\mu$ .

[2]

Use the formula $s_{n-1}^{2}=\frac{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}}{n-1}$ to find an unbiased estimate for the population variance.

[3]

Construct a $90 \%$ confidence interval for $\mu$ .

[3]

State a conclusion about a claim that the mean daily sales are 240 dollars, based on the confidence interval.

[2]

Question 7

HLPaper 1

A survey of 12 workers records their daily commute times (in minutes): $20,22,25,27$ , $30,32,35,38,40,42,45,50$ .

Calculate the $99 \%$ confidence interval for the population mean, $\mu$ , assuming the population standard deviation is unknown.

[4]

Sketch the confidence interval on a number line.

[2]

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

[1]

Question 8

HLPaper 1

A sample of 10 students' exam scores is: $60,62,65,68,70,72,75,78,80,82$ .

Find the $90 \%$ confidence interval for the population mean, $\mu$ , using the sample standard deviation.

[4]

If the population standard deviation is known to be $\sigma=7$ , find the $90 \%$ confidence interval.

[2]

Compare the widths of the confidence intervals in parts (a) and (b).

[2]

Question 9

HLPaper 1

A random sample of 40 cars is tested for fuel efficiency (in km/L), yielding a sample mean of $15.5 \mathrm{~km} / \mathrm{L}$ and a sample variance of $s_{n-1}^{2}=4$ . Assume the fuel efficiencies are normally distributed.

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

[3]

Sketch the confidence interval on a number line.

[3]

The manufacturer claims the mean fuel efficiency is $16 \mathrm{~km} / \mathrm{L}$ . Based on the confidence interval, comment on the validity of this claim.

[2]

State the Central Limit Theorem as it applies to this context.

[1]

Question 10

HLPaper 2

A researcher measures the height (in cm ) of a random sample of 12 trees, with the following summary statistics: $\sum h=1800, \sum h^{2}=270500$ , where $h$ is the height of a tree. Assume the heights are normally distributed.

Calculate unbiased estimates for the mean and variance of the tree heights.

[3]

Find the $95 \%$ confidence interval for the population mean height, $\mu$ .

[3]

A previous study suggested the mean height was 145 cm . Conduct a one-tailed hypothesis test at the $10 \%$ significance level to determine if the mean height is greater than 145 cm . State the hypotheses, test statistic, and conclusion.

[4]

AHL 4.16—Confidence intervals Questionbank