A $6$-sided die is rolled $120$ times. The observed frequencies for the scores $1$ to $6$ are $27, 12, 16, 25, 26,$ and $14$, respectively. Conduct a $\chi^2$ goodness of fit test at the $5\%$ significance level to determine if the die is fair. State your null hypothesis, the $p$-value, and your conclusion.

1. State the null hypothesis ($H_0$): The die is fair (the outcomes follow a uniform distribution). 2. Calculate the expected frequency for each face if the die is fair: $\frac{120}{6} = 20$. 3. Enter the observed list $\{27, 12, 16, 25, 26, 14\}$ and the expected list $\{20, 20, 20, 20, 20, 20\}$ into the GDC. 4. Execute the $\chi^2$ Goodness of Fit test on the GDC with $\text{df} = 6 - 1 = 5$. 5. Extract the $p$-value from the GDC output: $p = 0.0458$ (to 3 s.f.). 6. Compare the $p$-value to the significance level: $0.0458 < 0.05$. 7. Conclude : Since the $p$-value is less than $0.05$, we reject $H_0$. There is sufficient evidence to suggest the die is biased.

A factory machine produces bolts. The length of the bolts, $X$, is modelled by a normal distribution. The machine is working correctly if the mean length is $\mu = 50\text{ mm}$ with a standard deviation $\sigma = 1.2\text{ mm}$ ($H_0$). The machine is shut down for repairs if a randomly selected bolt has a length strictly greater than $52.5\text{ mm}$. Calculate the exact probability of making a Type I error.

1. Recall the definition of a Type I error: rejecting the null hypothesis $H_0$ when it is actually true. 2. Identify the rejection region from the prompt: $X > 52.5\text{ mm}$. 3. Set up the probability statement assuming $H_0$ is true: $P(X > 52.5 \mid \mu = 50, \sigma = 1.2)$. 4. Use the Normal Cumulative Distribution function on the GDC with lower bound $52.5$, upper bound $10^{99}$, $\mu = 50$, and $\sigma = 1.2$. 5. Evaluate the probability: $0.018610\dots$ 6. The probability of a Type I error is $0.0186$ (or $1.86\%$) .

Chi-Square Tests and Error Types Questions

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 chi-square goodness of fit, independence and error types

The question bank covers these chi-square goodness of fit, independence and error types question types (number of questions in brackets):

Chi-Square Goodness of Fit (10)
Type I and II Errors (7)
Chi-Square Test for Independence (3)

Key formulas

Chi-squared test statistic: $χ_{calc}^{2} = \sum \frac{(f_{o} - f_{e})^{2}}{f_{e}}$

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

Chi-Square Goodness of Fit, Independence and Error Types worked examples

Worked example 1: Calculating Expected Frequencies and Degrees of Freedom · easy

A researcher conducts a $χ^{2}$ test for independence at the $5 %$ significance level to determine if there is an association between a person's gender (Male, Female) and their preferred professional field (Engineering, Medicine, Law). The survey collected responses from $220$ individuals. In the sample, there were $110$ Males, and $90$ people in total preferred Engineering. Calculate the expected frequency of Males who prefer Engineering, and state the degrees of freedom for this test.

Solution

5 steps

Examiner tip: Always show your unrounded expected frequencies in your working if asked to construct an expected frequency table, as prematurely rounding these values can throw off your final

χ^{2}

statistic.

Worked example 2: Performing a $χ^{2}$ Goodness of Fit Test · medium

A $6$ -sided die is rolled $120$ times. The observed frequencies for the scores $1$ to $6$ are $27, 12, 16, 25, 26,$ and $14$ , respectively. Conduct a $χ^{2}$ goodness of fit test at the $5 %$ significance level to determine if the die is fair. State your null hypothesis, the $p$ -value, and your conclusion.

Solution

7 steps · 1 to fill in

Examiner tip: To score the final reasoning and accuracy marks in hypothesis testing, you must explicitly show the comparison of your

p

-value against the given significance level (e.g.,

0.0458 < 0.05

) before stating your conclusion in context.

Worked example 3: Calculating the Probability of a Type I Error · hard

A factory machine produces bolts. The length of the bolts, $X$ , is modelled by a normal distribution. The machine is working correctly if the mean length is $μ = 50 mm$ with a standard deviation $σ = 1.2 mm$ ( $H_{0}$ ). The machine is shut down for repairs if a randomly selected bolt has a length strictly greater than $52.5 mm$ . Calculate the exact probability of making a Type I error.

Solution

6 steps · 1 to fill in

Examiner tip: A Type I error (

α

) is calculated entirely using the parameters of the Null Hypothesis (

H_{0}

), whereas a Type II error (

β

) is calculated using the parameters of the Alternative Hypothesis (

H_{1}

).

Try these IB Maths AI HL chi-square goodness of fit, independence and error types 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 researcher conducts a Chi-Square test for independence to see if eye color (Blue, Green, Brown, Hazel) is independent of hair color (Blonde, Brown, Black, Red, Other).
Calculate the exact number of degrees of freedom ( $d f$ ) for this test.

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Question 2 · medium · 6 marks · Paper 1

A geneticist expects a ratio of $9 : 3 : 3 : 1$ for four phenotypes (A, B, C, D) in a certain plant. She grows $160$ plants and observes the following: - Phenotype A: $85$ - Phenotype B: $34$ - Phenotype C: $35$ - Phenotype D: $6$
Perform a Goodness of Fit test at the $10 %$ significance level to determine if the plants follow the expected genetic ratio.

Full worked solution Free, no sign-in

$H_{0}$ : the phenotypes occur in the ratio $9 : 3 : 3 : 1$ ; $H_{1}$ : they do not. [A1]
Ratio $9 : 3 : 3 : 1$ has $16$ parts in total. One part $= 160 / 16 = 10$ . [M1]
Expected frequencies: $A = 90, B = 30, C = 30, D = 10$ (all $\geq 5$ ). [A1]
Enter $f_{o}$ in List 1 ( $85, 34, 35, 6$ ) and $f_{e}$ in List 2 ( $90, 30, 30, 10$ ); $d f = 4 - 1 = 3$ . Run the $χ^{2}$ GOF test.
GDC outputs: $χ^{2} = 3.24$ ( $3.2444 . . .$ ) [A1]
$p$ -value $= 0.355$ ( $0.35545 . . .$ ) [A1]
Since $p = 0.355 > 0.10$ ( $α$ ), we do not reject $H_{0}$ : there is insufficient evidence to suggest the plants do not follow the $9 : 3 : 3 : 1$ ratio. [R1]

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

Question 3 · hard · 8 marks · Paper 1

A researcher claims that the heights of plants in a greenhouse follow a Normal distribution with $μ = 120$ cm and $σ = 15$ cm. He categorizes $200$ plants into three intervals: - Group 1: $< 105$ cm - Group 2: $105$ cm to $135$ cm - Group 3: $> 135$ cm
Calculate the expected frequency ( $f_{e}$ ) for each of the three groups, correct to 1 decimal place.

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Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 197 KB

All 20 chi-square goodness of fit, independence and error types questions with mark schemes →

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FAQ

How many IB Maths AI HL chi-square goodness of fit, independence and error types questions are there?

There are 20 exam-style chi-square goodness of fit, independence and error types questions in the AI HL question bank (Paper 1: 19 · Paper 2: 1), graded 9 easy, 6 medium, 5 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is chi-square goodness of fit, independence and error types on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 1. 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.

IB Maths AI HL Chi-Square Goodness of Fit, Independence and Error Types Questions: goodness of fit, independence and type I and II errors

Try it first

What you need to know

What's examined in AI HL chi-square goodness of fit, independence and error types

Key formulas

Chi-Square Goodness of Fit, Independence and Error Types worked examples

Worked example 1: Calculating Expected Frequencies and Degrees of Freedom · easy

Solution

Worked example 2: Performing a $χ^{2}$ Goodness of Fit Test · medium

Solution

Worked example 3: Calculating the Probability of a Type I Error · hard

Solution

Try these IB Maths AI HL chi-square goodness of fit, independence and error types questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 8 marks · Paper 1

Revise chi-square goodness of fit, independence and error types for the exam

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What else is free here

More AI HL Unit 4 topics

IB Maths AI HL Chi-Square Goodness of Fit, Independence and Error Types Questions: goodness of fit, independence and type I and II errors

Try it first

What you need to know

What's examined in AI HL chi-square goodness of fit, independence and error types

Key formulas

Chi-Square Goodness of Fit, Independence and Error Types worked examples

Worked example 1: Calculating Expected Frequencies and Degrees of Freedom · easy

Solution

Worked example 2: Performing a χ2 Goodness of Fit Test · medium

Solution

Worked example 3: Calculating the Probability of a Type I Error · hard

Solution

Try these IB Maths AI HL chi-square goodness of fit, independence and error types questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 8 marks · Paper 1

Revise chi-square goodness of fit, independence and error types for the exam

FAQ

What else is free here

More AI HL Unit 4 topics

Worked example 2: Performing a $χ^{2}$ Goodness of Fit Test · medium