IB Maths AI HL · Unit 4: Statistics and Probability
IB Maths AI HL Confidence Intervals and Hypothesis Testing for Means Questions
Exam-style IB Maths AI HL confidence intervals and hypothesis testing for means questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 17 questions
- Paper 1: 17
- 5 easy
- 5 medium
- 5 hard
- 2 starter
- 3 worked examples
Practise Confidence Intervals and Hypothesis Testing for Means questions →
AI HL formula booklet
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 (6)
- CI & Test Fundamentals (6)
- Two-sample Mean Inference (5)
Key formulas
- Confidence interval for a mean (large n)
- \(\bar{x} \pm z^* \cdot \dfrac{\sigma}{\sqrt{n}}\)
In the same notation as the IB formula booklet. All AI HL formulas →
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 ($\mu_b$) 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.
Examiner tip: When defining hypotheses for means, you must explicitly use the word "population" (or the symbol $\mu$). 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.
1. State the hypotheses: $H_0: \mu = 1000$ and $H_1: \mu < 1000$.
2. Navigate to the 1-Sample $t$-test tool on the GDC (using summary statistics, not raw data).
3. Input the parameters: $\mu_0 = 1000$, $\bar{x} = 980$, $s_x = 35$, $n = 15$, and select the alternative hypothesis form $\mu < \mu_0$.
4. Execute the test and extract the $p$-value: $0.02221\dots$
5. Compare with the significance level: $0.0222 < 0.05$.
6. Conclude: Reject $H_0$. The $p$-value is $0.0222$, providing sufficient evidence to conclude the mean lifespan is less than $1000$ hours.
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 ($\sigma_x$) 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 $215\text{ cm}$ with an unbiased sample standard deviation of $12\text{ cm}$. Construct a $95\%$ confidence interval for the population mean height of the sunflowers, and interpret its meaning.
1. Identify the correct GDC function: 1-Sample $t$-Interval (since the population standard deviation is unknown and estimated from the sample).
2. Input the summary statistics: Confidence Level $= 0.95$, $\bar{x} = 215$, $s_x = 12$, $n = 40$.
3. Execute the interval calculation on the GDC.
4. Extract the lower and upper bounds: $(211.16\dots, 218.83\dots)$.
5. State the interval to 3 significant figures: $[211\text{ cm}, 219\text{ cm}]$.
6. Interpret the interval: We are $95\%$ confident that the true population mean height of all fully grown sunflowers lies between $211\text{ cm}$ and $219\text{ cm}$.
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.
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.
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):
\[980, \quad 1010, \quad 995, \quad 975, \quad 985, \quad 1005, \quad 990, \quad 970\]
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.
Question 3 · hard · 7 marks · Paper 1
A \(90\%\) confidence interval for a population mean \(\mu\) was calculated using a sample of \(n=25\) observations. The population standard deviation \(\sigma\) was unknown, so the \(t\)-distribution was used.
The calculated interval is: \(120.4 < \mu < 129.6\).
Determine the exact sample standard deviation (\(s_{n-1}\)) that was calculated from the raw data to generate this interval.
All 17 confidence intervals and hypothesis testing for means questions with mark schemes →
FAQ
How many IB Maths AI HL confidence intervals and hypothesis testing for means questions are there?
There are 17 exam-style confidence intervals and hypothesis testing for means questions in the AI HL question bank (Paper 1: 17), graded 5 easy, 5 medium, 5 hard, 2 starter. 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?
In the question bank these questions are set as Paper 1 questions.
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.
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