DP Maths AI HL 4.11 Inferential Statistics

DP Maths AI HL 4.11 Inferential Statistics - Full notes

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DP Maths AI HL | 4.11 Inferential Statistics

DP MATHEMATICS AI · TOPIC 4: STATISTICS AND PROBABILITY

4.11

Inferential Statistics

Full notes for Applications and Interpretation HL

Contents

1	Why inferential statistics?
2	Null and alternative hypotheses
3	Significance levels and p-values
4	Contingency tables
5	Expected frequencies
6	The chi-squared statistic
7	Degrees of freedom and critical values
8	The chi-squared test for independence
9	Conditions and combining categories
10	Goodness of fit: given proportions
11	Goodness of fit: testing a distribution
12	Comparing two means: the t-test
13	One-tailed and two-tailed tests
14	Carrying out a t-test with technology
15	Interpreting results in context
16	Designing a fair study
17	Choosing the right test
18	Mixed review: worked problems
19	Confidence intervals (HL)
20	Errors in hypothesis testing and one-sample tests (HL)
21	Investigation: does music affect concentration?
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
write null and alternative hypotheses
interpret significance levels and p-values
calculate expected frequencies in contingency tables
carry out χ2 tests for independence with technology
carry out χ2 goodness of fit tests
carry out two-sample t-tests (one- and two-tailed)
state conclusions in context, and check conditions
choose an appropriate test

HL booklet: this booklet contains all the SL content plus confidence intervals, one-sample tests and Type I and II errors.

Guiding questions

How can sample data be used to make decisions about a population? How strong must the evidence be before we reject a claim?

How this topic is examined

Hypothesis tests are tested every session on Paper 2: write hypotheses, find expected values and degrees of freedom, obtain the χ2 statistic or t-test p-value with the GDC, and conclude in context. Paper 3 often contains an extended investigation using these tests.

Always compare the p-value with the significance level (or the statistic with the critical value), and write the conclusion in the words of the question, not just 'reject H0'.

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Why inferential statistics?

From sample to population.

Think about it

In a sample of 200 people, 58% of women and 49% of men preferred tea to coffee. Does this show that preference depends on gender, or could the difference be due to chance?

Descriptive statistics summarize data. Inferential statistics uses sample data to make decisions about a population, while accounting for random variation between samples. A hypothesis test asks: if there were no real effect, how likely is a result at least as extreme as the one observed? If that probability is very small, we have evidence of a real effect.

2 Null and alternative hypotheses

What are we testing?

The null hypothesis H0 states that there is no effect, no difference or no association (the 'status quo'). The alternative hypothesis H1 states what we are looking for evidence of. We assume H0 is true and see whether the data contradict it.

Test	H0	H1
χ2 independence	the variables are independent	the variables are not independent
χ2 goodness of fit	the data follow the stated distribution	the data do not follow it
t-test	μ1 = μ2	μ1 ≠ μ2, or μ1 > μ2, or μ1 < μ2

Practice exercise 1

1. Write hypotheses to test whether the choice of sport is independent of age group.

2. Write hypotheses to test whether a die is fair.

3 Significance levels and p-values

How strong is the evidence?

The significance level (usually 5% or 1%) is the threshold probability for rejecting H0. The p-value is the probability of obtaining a result at least as extreme as the observed one, assuming H0 is true. Decision rule: if p-value < significance level, reject H0; otherwise, do not reject H0 (there is insufficient evidence).

Common mistake

Saying 'H0 is true' when it is not rejected. We only conclude there is insufficient evidence to reject it.

Practice exercise 2

1. A test gives p = 0.031. What is the conclusion at the 5% and 1% levels?

4 Contingency tables

Two categorical variables.

A contingency table (two-way table) shows observed frequencies for combinations of two categorical variables. The row and column totals are used to calculate expected frequencies.

	Tea	Coffee	Neither	Total
Women	58	32	10	100
Men	49	41	10	100
Total	107	73	20	200

5 Expected frequencies

If the variables were independent.

WORKED EXAMPLE 1 Expected frequency (basic)

Find the expected number of women who prefer tea, assuming independence.

Solution

100 × 107/200M1

= 53.5A1

Practice exercise 3

1. Find the expected number of men who prefer coffee.

2. Find all expected frequencies for the 'Neither' column.

6 The chi-squared statistic

Measuring the difference.

fo is the observed and fe the expected frequency. A large χ2 means the observed data differ a lot from what independence predicts. The GDC calculates χ2 and the p-value directly from the observed matrix.

WORKED EXAMPLE 2 Calculating the chi-squared statistic (standard)

Find χ2 for the tea/coffee table.

Solution

Expected: 53.5, 36.5, 10 / 53.5, 36.5, 10A1

χ2 = 2(4.52/53.5) + 2(4.52/36.5) + 0M1

= 1.866A1

7 Degrees of freedom and critical values

How many cells are free?

For a goodness of fit test, ν = number of categories − 1 (minus one more for each parameter estimated from the data). The critical value is the value of χ2 beyond which we reject H0 at the given level; it is found from tables or the GDC (inverse χ2).

Figure 1. The χ2 distribution with 2 degrees of freedom; the shaded 5% tail is the rejection region.

Practice exercise 4

1. Find ν for a 3 × 4 table.

2. Find the 5% critical value for ν = 2.

8 The chi-squared test for independence

The full procedure.

1. State H0 (independent) and H1 (not independent). 2. Find expected frequencies (check all ≥ 5). 3. Find ν. 4. Use the GDC to find χ2 and the p-value. 5. Compare with the significance level (or critical value). 6. Conclude in context.

WORKED EXAMPLE 3 Independence test (exam level)

Test at 5% whether drink preference is independent of gender.

Solution

H0: preference and gender are independent; H1: not independentA1

ν = 2; χ2 = 1.866, p = 0.393A1 A1

p > 0.05: do not reject H0; insufficient evidence that preference depends on genderR1 A1

GDC tip

Enter the observed table as matrix [A]; run χ2-Test with observed [A], expected [B]. The GDC stores expected values in [B] and reports χ2, p and df.

Practice exercise 5

1. A 2 × 3 table gives χ2 = 7.42. Test at 5%.

2. Explain why the test cannot show that one variable causes the other.

9 Conditions and combining categories

Expected frequencies at least 5.

The χ2 test is only valid if all expected frequencies are at least 5. If some are smaller, combine adjacent rows or columns (sensibly), recalculate, and reduce the degrees of freedom accordingly. The data must be frequencies (counts), not percentages.

Practice exercise 6

1. A table has an expected frequency of 3.2 in one cell. What should be done?

10 Goodness of fit: given proportions

Does data match a claim?

A goodness of fit test compares observed frequencies with frequencies expected from a claimed distribution. Expected = total × claimed proportion. ν = number of categories − 1.

Score	1	2	3	4	5	6
Observed	18	22	15	25	14	26

WORKED EXAMPLE 4 Fair die (exam level)

A die is rolled 120 times (table). Test at 5% whether it is fair.

Solution

H0: fair; expected 20 eachA1

χ2 = (4 + 4 + 25 + 25 + 36 + 36)/20 = 6.5; ν = 5M1 A1

p = 0.261 > 0.05: do not reject H0; no evidence the die is biasedA1 R1

Practice exercise 7

1. A company claims its sweets are 30% red, 50% yellow, 20% green. A bag of 200 has 72 red, 92 yellow, 36 green. Test at 5%.

11 Goodness of fit: testing a distribution

Binomial or normal models.

Expected frequencies can come from a model such as B(n, p) or N(μ, σ2). If a parameter is estimated from the data, subtract one more degree of freedom for each estimated parameter (HL emphasis).

WORKED EXAMPLE 5 Binomial model (exam level)

For 100 families of 3 children, the number of girls was 0: 10, 1: 40, 2: 38, 3: 12. Test at 5% whether X ~ B(3, 0.5).

Solution

Expected 12.5, 37.5, 37.5, 12.5A1

χ2 = 0.5 + 0.167 + 0.0067 + 0.02 = 0.693; ν = 3M1 A1

p = 0.875: do not reject: consistent with B(3, 0.5)A1 R1

12 Comparing two means: the t-test

Is there a difference between groups?

The two-sample t-test tests whether two population means differ, using sample data. In AI it assumes both populations are normally distributed with equal variances (pooled test), and the samples are independent. H0: μ1 = μ2.

Method A scores	62, 70, 75, 68, 81, 73, 66, 77
Method B scores	58, 65, 60, 72, 63, 69, 61, 64

13 One-tailed and two-tailed tests

Direction matters.

A two-tailed test has H1: μ1 ≠ μ2 (a difference in either direction). A one-tailed test has H1: μ1 > μ2 or μ1 < μ2, used when we have a specific direction in mind before seeing the data. The p-value of a one-tailed test is half that of the two-tailed test (when the data go in the claimed direction).

14 Carrying out a t-test with technology

2-SampTTest.

WORKED EXAMPLE 6 t-test (exam level)

Test at 5% whether method A gives higher mean scores than method B.

Solution

H0: μA = μB; H1: μA > μBA1

GDC pooled 2-sample t-test: t = 2.73, p = 0.00810A1 A1

p < 0.05: reject H0; evidence that method A gives higher mean scoresR1 A1

GDC tip

Enter the samples in L1 and L2, run 2-SampTTest with Pooled: Yes and the correct direction for H1. Record t, p and the sample means.

Practice exercise 8

1. Redo the test as two-tailed. What is the p-value?

2. State two assumptions of the pooled t-test.

15 Interpreting results in context

Words, not just symbols.

A complete conclusion: (1) compares the p-value with the significance level, (2) states the decision about H0, (3) explains what this means in context. 'There is sufficient evidence at the 5% level to conclude that method A gives a higher mean score' is a good conclusion.

16 Designing a fair study

Collecting data for a test.

Valid inference needs random samples, sufficient sample sizes (expected frequencies ≥ 5 for χ2), independence between individuals, and measurement that is consistent. Confounding variables can produce associations that are not causal.

Practice exercise 9

1. A student surveys their friends about study habits and grades. Identify two weaknesses.

17 Choosing the right test

Match the data to the test.

Question	Data	Test
Are two categorical variables associated?	counts in a table	χ2 independence
Does data fit a claimed distribution?	counts in categories	χ2 goodness of fit
Do two groups have different means?	measurements in two groups	t-test

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 7 Mixed problem (exam level)

A 3 × 3 contingency table gives χ2 = 10.2. Test at 1%.

Solution

ν = 4; critical value 13.28A1 A1

10.2 < 13.28: do not reject H0 at 1%R1

Practice exercise 10

1. Find the p-value for χ2 = 10.2 with ν = 4, and the conclusion at 5%.

19 Confidence intervals (HL)

A range of plausible values.

A confidence interval for a population mean is x̄ ± zσ/√n (known σ) or x̄ ± t sn−1/√n (unknown σ). A 95% confidence interval means that 95% of such intervals, from repeated samples, would contain the true mean.

WORKED EXAMPLE 8 Confidence interval (exam level)

Method A scores (n = 8) have mean 71.5 and sn−1 = 6.21. Find a 95% confidence interval for the mean.

Solution

t-interval: 71.5 ± 2.365(6.21/√8)M1

(66.3, 76.7)A1

Practice exercise 11

1. How does increasing n affect the width of a confidence interval?

20 Errors in hypothesis testing and one-sample tests (HL)

Type I and Type II.

A Type I error rejects a true H0; its probability is the significance level. A Type II error fails to reject a false H0. One-sample tests compare a sample mean with a claimed value μ0 (z-test if σ known, t-test otherwise), or a count with a binomial or Poisson model.

WORKED EXAMPLE 9 One-sample t-test (exam level)

A manufacturer claims bulbs last 1000 hours. A sample of 10 has mean 968 h and sn−1 = 45 h. Test at 5% whether the mean is less than 1000.

Solution

t = (968 − 1000)/(45/√10) = −2.249M1 A1

p = 0.0255 < 0.05: reject H0; evidence that bulbs last less than claimedA1 R1

Practice exercise 12

1. Explain the consequence of a Type I error in the bulb example.

21 Investigation: does music affect concentration?

A Paper 3 style exploration.

Students complete a puzzle either in silence (group S) or with music (group M). Times (s): S: 48, 52, 45, 50, 55, 47, 51, 49; M: 54, 58, 50, 61, 55, 57, 53, 60. 1. Find the mean and standard deviation of each group. 2. Write hypotheses to test whether music increases mean time. 3. Carry out a pooled t-test at 5%. 4. State your conclusion in context. 5. Discuss whether the study design supports a causal conclusion.

Answers. 1. S: mean 49.6, SD 2.9; M: mean 56.0, SD 3.5 (population SDs). 2. H0: μM = μS; H1: μM > μS. 3. t ≈ 3.73, p ≈ 0.001. 4. Reject H0: evidence that music increases the mean time. 5. Only if students were randomly assigned to groups; otherwise confounding (ability, tiredness) is possible.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 2 style) (about 15 minutes)

A survey of 300 students records preferred learning style by year group: Year 10: visual 40, auditory 25, kinaesthetic 35; Year 11: visual 45, auditory 30, kinaesthetic 25; Year 12: visual 55, auditory 20, kinaesthetic 25.

(a) State H0 and H1. [1]

(b) Find the expected number of Year 12 visual learners. [2]

(d) Find the χ2 statistic and p-value. [2]

(e) State the conclusion at 5%. [2]

Markscheme

Part	Answer	Marks
(a)	H0: learning style independent of year group; H1: not independent	A1
(b)	100 × 140/300	M1
	46.7	A1
(c)	4	A1
(d)	χ2 = 6.85	A1
	p = 0.144	A1
(e)	p > 0.05	R1
	insufficient evidence of association	A1

Examiner's comments. Conclusions must be in context.

Question 2 (Paper 2 style) (about 12 minutes)

Two fertilizers are tested on plants. Heights (cm): F: 24, 27, 22, 29, 26, 25; G: 21, 23, 20, 24, 22, 25.

(a) State hypotheses for a two-tailed test. [1]

(b) Carry out a pooled t-test and find the p-value. [2]

Markscheme

Part	Answer	Marks
(a)	H0: μF = μG; H1: μF ≠ μG	A1
(b)	t = 2.40	A1
	p = 0.0375	A1
(c)	p < 0.05: reject H0	R1
	evidence that mean heights differ	A1

Examiner's comments. Use the pooled option.

Question 3 (Paper 1 style) (about 6 minutes)

A goodness of fit test with 5 categories gives χ2 = 11.3. Test at 5% using the critical value 9.488. [3]

Markscheme

Part	Answer	Marks
	ν = 4	A1
	11.3 > 9.488	R1
	reject H0: the data do not fit the claimed distribution	A1

Examiner's comments. State the degrees of freedom.

23 Common misconceptions and links

Students often think…	Actually…
not rejecting H0 proves it	it means insufficient evidence
χ2 tests use percentages	use frequencies
association means causation	only association
any expected frequency is fine	all should be ≥ 5

How 4.11 connects to the rest of the course

Topic	Connection
4.2 Statistics	means and SD
4.5 Probability	independence
4.7 Distributions	binomial and normal models
4.8 Binomial	expected frequencies

24 Check your understanding

Section A: multiple choice

1. p = 0.02 at the 5% level means

A. reject H0 B. do not reject H0 C. H0 is true D. H1 is false

2. ν for a 2 × 4 table is

A. 8 B. 6 C. 3 D. 4

3. Expected frequency =

A. row total/column total B. row total × column total / grand total C. observed/total D. grand total/cells

4. A t-test compares

A. two proportions B. counts C. distributions D. two means

5. Goodness of fit ν with 6 categories and no estimated parameters:

A. 6 B. 5 C. 4 D. 1

Section B: structured

6. Find the expected frequency for a cell with row total 60, column total 45, grand total 180. [2]

7. Explain the meaning of a 5% significance level. [1]

Answers

1. A 2. C 3. B 4. D 5. B

6. 15

7. the probability of rejecting H0 when it is true is 5%

25 Summary and topic map

Idea	Remember
Hypotheses	H0: no effect; H1: effect
p-value	reject H0 if p < significance level
Expected	row × column/total
χ2 independence	ν = (r − 1)(c − 1)
Goodness of fit	ν = categories − 1 (− parameters)
t-test	two means; pooled; one- or two-tailed
Conclusion	in context

Before the exam, make sure you…

•check expected frequencies ≥ 5
•state the direction of H1
•compare p with α
•conclude in context

Inferential Statistics on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 2. Topic map for Inferential Statistics.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. H0: sport and age group are independent; H1: they are not independent

2. H0: each face has probability 1/6; H1: the die is not fair

Practice exercise 2

1. reject H0 at 5%; do not reject at 1%

Practice exercise 3

1. 36.5

2. 10 and 10

Practice exercise 4

1. 6

2. 5.991

Practice exercise 5

1. ν = 2, critical value 5.991: reject H0

2. the test shows association only

Practice exercise 6

1. combine that category with a neighbouring one and redo the test

Practice exercise 7

1. expected 60, 100, 40: χ2 = 3.44, ν = 2, p = 0.179: do not reject

Practice exercise 8

1. p = 0.0162: still reject at 5%

2. normal populations; equal variances (and independent random samples)

Practice exercise 9

1. convenience sample (biased); small sample; friends may not be independent

Practice exercise 10

1. p = 0.0372: reject at 5%

Practice exercise 11

1. it becomes narrower (width ∝ 1/√n)

Practice exercise 12

1. concluding the bulbs are worse than claimed when they are not

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