Arriane claims that the population mean weight of eggs from black geese ($\mu_b$) is greater than the population mean weight of eggs from white geese ($\mu_w$). A two-sample $t$-test is performed at the $10\%$ significance level. The GDC outputs a $p$-value of $0.177$. Write down the null hypothesis, and state whether Arriane's claim is supported by the evidence.

1. State the null hypothesis ($H_0$) representing no difference: $H_0: \mu_b = \mu_w$ (The population mean weight of eggs from black geese is equal to that of white geese). 2. Identify the $p$-value: $0.177$. 3. Compare the $p$-value to the significance level ($10\% = 0.10$): $0.177 > 0.10$. 4. Determine the outcome: Since the $p$-value is greater than the significance level, there is insufficient evidence to reject $H_0$. 5. Conclude in context: Arriane's claim is not supported by the evidence.

A factory produces lightbulbs that are supposed to last for an average of $1000$ hours. A quality control manager tests a random sample of $8$ bulbs and records their lifespans (in hours): \[980, \quad 1010, \quad 995, \quad 975, \quad 985, \quad 1005, \quad 990, \quad 970\] Assuming the lifespans are normally distributed, perform a $1$ -sample $t$ -test at the $10\%$ significance level to determine if the true mean lifespan is less than $1000$ hours. State your $p$ -value and conclusion.

$H_0: \mu = 1000$ , $H_1: \mu < 1000$ (left-tailed test). [A1] Enter the data into List 1 ( $\bar{x} = 988.75$ , $s_{n-1} = 14.08$ ). Using GDC: STAT $\rightarrow$ TEST $\rightarrow$ t $\rightarrow$ 1-Sample , with Data $=$ List, $\mu < \mu_0$ , $\mu_0 = 1000$ . [M1] GDC outputs: $t = -2.26$ ( $-2.2601...$ ) [A1] $p$ -value $= 0.0292$ ( $0.029160...$ ) [A1] Since $p = 0.0292 < 0.10$ ( $\alpha$ ), we reject $H_0$ . [R1] There is sufficient evidence that the true mean lifespan is less than $1000$ hours.

Confidence Intervals and Hypothesis Testing for Means Questions

Q: How many IB Maths AI HL confidence intervals and hypothesis testing for means questions are there?

There are 25 exam-style confidence intervals and hypothesis testing for means questions in the AI HL question bank (Paper 1: 19 · Paper 2: 6), graded 8 easy, 9 medium, 7 hard, 1 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is confidence intervals and hypothesis testing for means on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 6. Practise with your GDC — AI papers expect calculator methods throughout.

Q: Where can I get the mark schemes?

Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What you need to know

HL AI extends SL AI's chi-squared with confidence intervals for the population mean. Practise the t-distribution CDF on your GDC. Chi-squared test and confidence intervals overview →

One-sample and two-sample t-tests when the population variance is unknown. HL AI expects the null, alternative, test statistic, and p-value each stated explicitly. Hypothesis testing — t-test and chi-squared overview →

What's examined in AI HL confidence intervals and hypothesis testing for means

The question bank covers these confidence intervals and hypothesis testing for means question types (number of questions in brackets):

One-sample Mean Inference (10)
CI & Test Fundamentals (10)
Two-sample Mean Inference (5)

Key formulas

Confidence interval for a mean (large n): $\bar{x} \pm z^{*} \cdot \frac{σ}{\sqrt{n}}$

In the same notation as the IB formula booklet. All AI HL formulas →

Where students lose marks on confidence intervals and hypothesis testing for means

Examiner's noteHypothesis tests: hypotheses in words, the wrong test, and conclusions that don't follow

Where marks are lost: All five sessions. Hypotheses written in words that mix up sample and population (or omit the population mean); a two-tailed alternative for a one-tailed test; a two-sample test where a paired test was needed (and its assumptions not stated); a z-test instead of a t-test; χ² goodness-of-fit degrees of freedom not reduced for an estimated parameter; cells with expected frequency below 5 not combined; the binomial test for a proportion hardly known; a correct comparison followed by the opposite conclusion; and confidence intervals without inequality notation.

What examiners look for: Write H₀ and H₁ with symbols for population parameters (μ, ρ, p). Choose the test from the design (paired data → paired t; σ unknown → t). State the p-value, compare it with the stated significance level, then conclude in context. For goodness of fit: df = cells − 1 − parameters estimated.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Confidence Intervals and Hypothesis Testing for Means worked examples

Worked example 1: Interpreting a Two-Sample t-test · easy

Arriane claims that the population mean weight of eggs from black geese ( $μ_{b}$ ) is greater than the population mean weight of eggs from white geese ( $μ_{w}$ ). A two-sample $t$ -test is performed at the $10 %$ significance level. The GDC outputs a $p$ -value of $0.177$ . Write down the null hypothesis, and state whether Arriane's claim is supported by the evidence.

Solution

5 steps

Examiner tip: When defining hypotheses for means, you must explicitly use the word "population" (or the symbol

μ

). Testing compares population parameters, not sample statistics (

\bar{x}

).

Worked example 2: Performing a 1-Sample t-test · medium

A manufacturer claims their lightbulbs have a mean lifespan of $1000$ hours. A consumer watchdog tests a random sample of $15$ lightbulbs and finds a sample mean of $980$ hours with a sample standard deviation of $35$ hours. Conduct a 1-tailed $t$ -test at the $5 %$ significance level to determine if the mean lifespan is less than the manufacturer's claim. State the $p$ -value and your conclusion.

Solution

6 steps

Examiner tip: Ensure you input the unbiased sample standard deviation (

s_{n - 1}

or

s_{x}

) into your GDC rather than the population standard deviation (

σ_{x}

) when performing

t

-tests from summary statistics.

Worked example 3: Constructing a Confidence Interval · hard

A botanist measures the heights of a random sample of $40$ fully grown sunflower plants. The sample mean height is $215 cm$ with an unbiased sample standard deviation of $12 cm$ . Construct a $95 %$ confidence interval for the population mean height of the sunflowers, and interpret its meaning.

Solution

6 steps · 1 to fill in

Examiner tip: A confidence interval estimates the population mean, not the range of individual values. Do not say "95\% of the sunflowers are between 211cm and 219cm."

Try these IB Maths AI HL confidence intervals and hypothesis testing for means questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

A school claims that its students study an average of $15$ hours per week. A skeptical researcher believes the true average is actually less than $15$ hours. They take a sample of $40$ students to test this.
State the null hypothesis ( $H_{0}$ ) and the alternative hypothesis ( $H_{1}$ ) for this test.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

A factory produces lightbulbs that are supposed to last for an average of $1000$ hours. A quality control manager tests a random sample of $8$ bulbs and records their lifespans (in hours): $980, 1010, 995, 975, 985, 1005, 990, 970$ Assuming the lifespans are normally distributed, perform a $1$ -sample $t$ -test at the $10 %$ significance level to determine if the true mean lifespan is less than $1000$ hours. State your $p$ -value and conclusion.

Full worked solution Free, no sign-in

$H_{0} : μ = 1000$ , $H_{1} : μ < 1000$ (left-tailed test). [A1]
Enter the data into List 1 ( $\bar{x} = 988.75$ , $s_{n - 1} = 14.08$ ).
Using GDC: STAT $\to$ TEST $\to$ t $\to$ 1-Sample, with Data $=$ List, $μ < μ_{0}$ , $μ_{0} = 1000$ . [M1]
GDC outputs: $t = - 2.26$ ( $- 2.2601 . . .$ ) [A1]
$p$ -value $= 0.0292$ ( $0.029160 . . .$ ) [A1]
Since $p = 0.0292 < 0.10$ ( $α$ ), we reject $H_{0}$ . [R1]
There is sufficient evidence that the true mean lifespan is less than $1000$ hours.

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 1

A $90 %$ confidence interval for a population mean $μ$ was calculated using a sample of $n = 25$ observations. The population standard deviation $σ$ was unknown, so the $t$ -distribution was used. The calculated interval is: $120.4 < μ < 129.6$ .
Determine the exact sample standard deviation ( $s_{n - 1}$ ) that was calculated from the raw data to generate this interval.

Attempt it and see the mark scheme →

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All 25 confidence intervals and hypothesis testing for means questions with mark schemes →

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FAQ

How many IB Maths AI HL confidence intervals and hypothesis testing for means questions are there?

There are 25 exam-style confidence intervals and hypothesis testing for means questions in the AI HL question bank (Paper 1: 19 · Paper 2: 6), graded 8 easy, 9 medium, 7 hard, 1 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is confidence intervals and hypothesis testing for means on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 6. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Confidence intervals for a mean (GDC).

IB Maths AI HL Confidence Intervals and Hypothesis Testing for Means Questions

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What you need to know

What's examined in AI HL confidence intervals and hypothesis testing for means

Key formulas

Where students lose marks on confidence intervals and hypothesis testing for means

Confidence Intervals and Hypothesis Testing for Means worked examples

Worked example 1: Interpreting a Two-Sample t-test · easy

Solution

Worked example 2: Performing a 1-Sample t-test · medium

Solution

Worked example 3: Constructing a Confidence Interval · hard

Solution

Try these IB Maths AI HL confidence intervals and hypothesis testing for means questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

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Question 3 · hard · 7 marks · Paper 1

Revise confidence intervals and hypothesis testing for means for the exam

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