Mini-Lesson

HL Statistics (AI HL)

3 missions. Guess, learn, then check. Your place is saved.

Mission 1A survey says students sleep 7.1 ± 0.2 hours. What does ± promise?
Mission 2Usually 3 complaints a week. This week, 8. Bad luck or a real problem?
Mission 3Two curves fit the same data. Which is better, and can you prove it?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

A survey says students sleep 7.1 ± 0.2 hours. What does ± promise?

Make a guess: A survey says students sleep 7.1 ± 0.2 hours. What does ± promise?

HL statistics · estimators and confidence intervals

Transforming and combining random variables

One variable: $E (a X + b) = a E (X) + b$ E(aX+b) = aE(X)+b and $Var (a X + b) = a^{2} Var (X)$ Var(aX+b) = a²Var(X). Adding $b$ b shifts; it never changes the spread.
Any $X$ X and $Y$ Y: $E (a X \pm bY) = a E (X) \pm b E (Y)$ E(aX ± bY) = aE(X) ± bE(Y).
Independent $X$ X and $Y$ Y: $Var (a X \pm bY) = a^{2} Var (X) + b^{2} Var (Y)$ Var(aX ± bY) = a²Var(X) + b²Var(Y). Variances always add, even for a difference.
Two separate observations are not the same as doubling one: $Var (X_{1} + X_{2}) = 2 Var (X)$ Var(X₁ + X₂) = 2Var(X) but $Var (2 X) = 4 Var (X)$ Var(2X) = 4Var(X).
A linear combination of independent normal variables is normal.

Worked example HL statistics · estimators and confidence intervals

Packing a box of flour

Question

Bags of flour have mass

X \sim N (1010, 6^{2})

grams. An empty box has mass

Y \sim N (250, 4^{2})

grams. A full box holds 5 bags, all masses independent. Find the probability that a full box weighs more than 5,320 g.

1
Write the total as separate bags: $T = X_{1} + X_{2} + X_{3} + X_{4} + X_{5} + Y$ T = X₁ + X₂ + X₃ + X₄ + X₅ + Y
2
Expected value: $E (T) = 5 \times 1010 + 250 = 5300$ E(T) = 5 × 1010 + 250 = 5300
3
Variance, adding each part: $Var (T) = 5 \times 36 + 16 = 196$ Var(T) = 5 × 36 + 16 = 196, so the standard deviation is 14
4
$T$ T is normal, so use your GDC: $P (T > 5320)$ P(T > 5320) with $T \sim N (5300, 1 4^{2})$ T ~ N(5300, 14²) gives 0.0766

Answer

0.0766 (3 s.f.)

⚠️ Watch out

writing

5 X

gives

Var (T) = 25 \times 36 + 16 = 916

and a probability of 0.254. Five different bags are

X_{1} + \dots + X_{5}

, not one bag counted five times.

Quick check

?

X

and

Y

are independent with

Var (X) = 4

and

Var (Y) = 1

. Find

Var (2 X - 3 Y)

.

HL statistics · estimators and confidence intervals

Estimators and the central limit theorem

$x ˉ = \frac{\sum x _{i}}{n}$ x̄ = (Σ xᵢ)/n is an unbiased estimate of the population mean $μ$ μ.
$s_{n - 1 2} = \frac{n}{n - 1} s_{n 2}$ sₙ₋₁² = n/(n−1)sₙ² is an unbiased estimate of $σ^{2}$ σ². Dividing by $n$ n tends to underestimate it.
If $X \sim N (μ, σ^{2})$ X ~ N(μ, σ²), then $X ˉ \sim N (μ, \frac{σ ^{2}}{n})$ X̄ ~ N(μ, σ²/n) for any $n$ n.
Central limit theorem: whatever the distribution of $X$ X, $X ˉ$ X̄ is approximately $N (μ, \frac{σ ^{2}}{n})$ N(μ, σ²/n) for large $n$ n. In IB exams, $n > 30$ n > 30 counts as large.

Calculate

?A random sample of 8 apples has masses, in grams: 48, 52, 50, 47, 55, 51, 49, 54. Use your GDC to find an unbiased estimate of the population variance. Give your answer to 3 significant figures.

Calculate

?The number of people in a car on a city road has mean 1.6 and standard deviation 0.9. Its distribution is not normal. A random sample of 50 cars is taken. Use the central limit theorem to estimate the probability that the sample mean is greater than 1.75. Give your answer to 3 significant figures.

HL statistics · estimators and confidence intervals

Confidence intervals for a mean

A confidence interval is estimate ± margin: $x ˉ \pm z^{*} \frac{σ}{n}$ x̄ ± z^(*)σ/(√n) when $σ$ σ is known.
When $σ$ σ is unknown, use $s_{n - 1}$ sₙ₋₁ and the $t$ t-distribution with $n - 1$ n−1 degrees of freedom, whatever the sample size. Your GDC does both.
Meaning: if you took many samples and built a 95% interval from each, about 95% of the intervals would contain $μ$ μ. The 95% belongs to the method, not to one interval.
Higher confidence or a smaller sample gives a wider interval.

Worked example HL statistics · estimators and confidence intervals

A $t$ t-interval for mean sleep

Question

A random sample of 12 students records sleep times, assumed normal. The sample has

x ˉ = 7.1

hours and

s_{n - 1} = 0.31

hours. Find a 95% confidence interval for the mean sleep time of all students.

1
$σ$ σ is unknown, so use a $t$ t-interval with $12 - 1 = 11$ 12 − 1 = 11 degrees of freedom
2
Use your GDC with $x ˉ = 7.1$ x̄ = 7.1, $s_{n - 1} = 0.31$ sₙ₋₁ = 0.31, $n = 12$ n = 12, level 0.95: the critical value is $t^{*} \approx 2.201$ t^(*) ≈ 2.201
3
Margin: $2.201 \times \frac{0.31}{12} \approx 0.197$ 2.201 × 0.31/(√12) ≈ 0.197
4
Interval: $7.1 \pm 0.197$ 7.1 ± 0.197

Answer

6.90 < μ < 7.30

hours (3 s.f.)

⚠️ Watch out

using

z^{*} = 1.96

here gives a narrower interval,

6.92

to

7.28

. With

σ

unknown the IB expects

t

, even for large samples.

Calculate

?A machine fills bottles. The volumes are normal with known standard deviation 2.4 ml. A sample of 36 bottles has mean 51.3 ml. Find the lower limit of a 99% confidence interval for the mean volume. Give your answer to 3 significant figures.

ml

Mission check

?A survey says students sleep 7.1 ± 0.2 hours (95% confidence). What does ± promise?

Mission 2

Usually 3 complaints a week. This week, 8. Bad luck or a real problem?

Make a guess: Usually 3 complaints a week. This week, 8. Bad luck or a real problem?

HL statistics · Poisson and hypothesis tests

The Poisson model

$X \sim Po (m)$ X ~ Po(m) counts events in a fixed interval: $P (X = x) = \frac{m ^{x} e ^{- m}}{x !}$ P(X = x) = (m^x e^(−m))/(x!), for $x = 0, 1, 2, \dots$ x = 0, 1, 2, …
Mean = variance = $m$ m.
Use it when events are independent and occur at a uniform average rate. Scale $m$ m with the interval: 3 a week is 6 a fortnight.
If $X \sim Po (m_{1})$ X ~ Po(m₁) and $Y \sim Po (m_{2})$ Y ~ Po(m₂) are independent, then $X + Y \sim Po (m_{1} + m_{2})$ X + Y ~ Po(m₁ + m₂).

Worked example HL statistics · Poisson and hypothesis tests

Two help desks

Question

Calls reach help desk A at an average rate of 2.5 per hour and desk B at 1.5 per hour, independently and at random. Find the probability that the two desks receive at most 2 calls in total in one hour.

1
Add the independent Poisson means: $T \sim Po (2.5 + 1.5) = Po (4)$ T ~ Po(2.5 + 1.5) = Po(4)
2
Write the event: $P (T \leq 2) = P (T = 0) + P (T = 1) + P (T = 2)$ P(T ≤ 2) = P(T = 0) + P(T = 1) + P(T = 2)
3
Substitute: $e^{- 4} (1 + 4 + \frac{4 ^{2}}{2}) = 13 e^{- 4}$ e⁻⁴(1 + 4 + 4²/2) = 13e⁻⁴
4
Evaluate, or use the Poisson cdf on your GDC with $m = 4$ m = 4, upper bound 2

Answer

13 e^{- 4} \approx 0.238

(3 s.f.)

⚠️ Watch out

"at most 2" includes 2. Using

P (T < 2)

leaves out

P (T = 2)

and gives 0.0916.

Sort it

?Choose the best model for each random variable.

Tap a statement, then tap the group it belongs in. Then press Check.

HL statistics · Poisson and hypothesis tests

Testing a parameter; critical regions and errors

Binomial and Poisson tests are one-tailed, e.g. $H_{0} : m = 3$ H₀: m = 3, $H_{1} : m > 3$ H₁: m > 3.
$p$ p-value: probability, under $H_{0}$ H₀, of a result at least as extreme as the one seen. Reject $H_{0}$ H₀ if $p <$ p < the significance level.
Critical region: the values that lead to rejecting $H_{0}$ H₀. For a discrete variable, make $P (X \in CR)$ P(X ∈ CR) as large as possible while staying below the significance level.
Type I error: rejecting a true $H_{0}$ H₀; its probability is $P (X \in CR ∣ H_{0})$ P(X ∈ CR | H₀). Type II error: accepting a false $H_{0}$ H₀.

Worked example HL statistics · Poisson and hypothesis tests

A binomial test for a proportion

Question

A seed company claims 80% of its seeds germinate. A gardener suspects the rate is lower. She plants 20 seeds and 12 germinate. Test the claim at the 5% significance level.

1
Hypotheses: $H_{0} : p = 0.8$ H₀: p = 0.8, $H_{1} : p < 0.8$ H₁: p < 0.8, where $p$ p is the proportion that germinate
2
Under $H_{0}$ H₀, $X \sim B (20, 0.8)$ X ~ B(20, 0.8)
3
Use your GDC: $p$ p-value $= P (X \leq 12) \approx 0.0321$ = P(X ≤ 12) ≈ 0.0321
4
Compare: $0.0321 < 0.05$ 0.0321 < 0.05, so reject $H_{0}$ H₀

Answer

there is significant evidence at the 5% level that fewer than 80% of the seeds germinate.

⚠️ Watch out

the

p

-value is

P (X \leq 12)

, not

P (X = 12) \approx 0.0222

. "At least as extreme" means 12 or fewer here.

Calculate

?A road has an average of 2.5 accidents a month. Residents think a new junction has increased this. The council tests

H_{0} : m = 2.5

against

H_{1} : m > 2.5

at the 5% level, using

X

, the number of accidents in one month. The critical region is

X \geq c

. Find

c

.

Calculate

?For the test on the previous screen (critical region

X \geq 6

), the true mean rate is in fact 5 accidents a month. Find the probability of a Type II error. Give your answer to 3 significant figures.

HL statistics · Poisson and hypothesis tests

Tests for a mean and for correlation

$σ$ σ known: $z$ z-test on $x ˉ$ x̄, with $X ˉ \sim N (μ_{0}, \frac{σ ^{2}}{n})$ X̄ ~ N(μ₀, σ²/n) under $H_{0}$ H₀. Critical values and error probabilities can be found.
$σ$ σ unknown: $t$ t-test using $s_{n - 1}$ sₙ₋₁, whatever the sample size. Use the $p$ p-value from your GDC.
Paired samples (before and after, on the same people): find each difference $d$ d, then do a one-sample test on the differences, $H_{0} : μ_{d} = 0$ H₀: μ_d = 0.
Correlation: for bivariate normal data, your GDC tests $H_{0} : ρ = 0$ H₀: ρ = 0 and gives a $p$ p-value.

Calculate

?Bags of rice have masses that are normal with known standard deviation 4 g. The label says the mean is 500 g. An inspector weighs a random sample of 16 bags and tests

H_{0} : μ = 500

against

H_{1} : μ < 500

at the 5% significance level. The true mean is in fact 497 g. Find the probability of a Type II error. Do not round the critical value, and give your answer to 3 significant figures.

Mission check

?Usually 3 complaints a week. This week, 8. Bad luck or a real problem?

Mission 3

Two curves fit the same data. Which is better, and can you prove it?

Make a guess: Two curves fit the same data. Which is better, and can you prove it?

HL statistics · data collection and non-linear regression

Designing valid data collection

Questions must be unbiased (not leading), precise and, where possible, structured with consistent answer choices. Open, unstructured questions are hard to analyse.
Avoid needlessly personal questions: people skip them or answer untruthfully.
Choose only the relevant variables and data for your question.
For a $χ^{2}$ chi² test, choose categories so every expected frequency is greater than 5, merging neighbouring classes if needed.
In a goodness of fit test, each parameter estimated from the data removes one more degree of freedom.

Sort it

?Sort each survey question.

Tap a statement, then tap the group it belongs in. Then press Check.

HL statistics · data collection and non-linear regression

Reliability and validity

Reliable: gives consistent results when repeated. Valid: measures what it claims to measure. A test can be reliable but not valid.
Test-retest (reliability): the same people take the same test again later; the scores should correlate strongly.
Parallel forms (reliability): the same people take two equivalent versions; the scores should correlate strongly.
Content validity: the test covers the full range of what it should measure.
Criterion-related validity: results agree with an established measure (the criterion).

Quick check

?A new 10-minute fitness test is given to 40 athletes, who also run a standard lab treadmill test that is known to be accurate. The two sets of scores have a correlation of 0.88. What has been checked?

Calculate

?The number of goals in each of 80 matches is recorded. A

χ^{2}

goodness of fit test checks whether a Poisson model fits, with the mean estimated from the data as 1.9. The categories are 0, 1, 2, 3, 4 and 5 or more, but the expected frequency for "5 or more" is about 3.5, so it is merged with "4". How many degrees of freedom does the test have?

HL statistics · data collection and non-linear regression

Residuals, $S S_{res}$ SS_res and $R^{2}$ R²

Your GDC fits least squares models: linear, quadratic, cubic, exponential, power and sine.
A residual is $y - y^$ y − ŷ: observed minus predicted. The sum of square residuals $S S_{res} = \sum (y - y^)^{2}$ SS_res = Σ (y − ŷ)². Smaller means a closer fit.
$R^{2}$ R² is the proportion of the variability in $y$ y explained by the model. For a linear model, $R^{2} = r^{2}$ R² = r².
$R^{2}$ R² alone can't choose a model: extra terms never lower it. Check the model makes sense in context.

Worked example HL statistics · data collection and non-linear regression

$S S_{res}$ SS_res for an exponential model

Question

An app's users

y

(thousands) after

x

months are

(0, 2.1)

,

(1, 2.9)

,

(2, 4.6)

,

(3, 6.8)

,

(4, 10.3)

. Model A is

y^= 2 \times 1. 5^{x}

. Find

S S_{res}

for model A.

1
Predicted values: $y^= 2, 3, 4.5, 6.75, 10.125$ ŷ = 2, 3, 4.5, 6.75, 10.125
2
Residuals $y - y^$ y − ŷ: $0.1, - 0.1, 0.1, 0.05, 0.175$ 0.1, −0.1, 0.1, 0.05, 0.175
3
Square: $0.01, 0.01, 0.01, 0.0025, 0.030625$ 0.01, 0.01, 0.01, 0.0025, 0.030625
4
Add: $S S_{res} = 0.063125$ SS_res = 0.063125

Answer

S S_{res} \approx 0.0631

(3 s.f.)

⚠️ Watch out

square each residual before adding. The residuals themselves add to 0.325, and positive and negative residuals can cancel.

Calculate

?For the same app data, model B is the straight line

y^= 2 x + 1.5

. Find

S S_{res}

for model B, exactly as a decimal.

Calculate

?A farmer records yield

y

(kg per plot) for fertiliser

x

(kg per plot):

(0, 1.5)

,

(1, 5.6)

,

(2, 9.3)

,

(3, 9.4)

,

(4, 9.0)

,

(5, 5.1)

. Use your GDC to fit a quadratic regression model and find the coefficient of determination

R^{2}

. Give your answer to 3 significant figures.

Mission check

?Two curves fit the same data. Which is better, and can you prove it?

Recap

Big ideas

$E (a X \pm bY) = a E (X) \pm b E (Y)$ E(aX ± bY) = aE(X) ± bE(Y); for independent variables $Var (a X \pm bY) = a^{2} Var (X) + b^{2} Var (Y)$ Var(aX ± bY) = a²Var(X) + b²Var(Y).
$x ˉ$ x̄ and $s_{n - 1 2}$ sₙ₋₁² are unbiased estimates; $X ˉ$ X̄ has standard deviation $\frac{σ}{n}$ σ/(√n) and is approximately normal for $n > 30$ n > 30.
Confidence interval for $μ$ μ: normal if $σ$ σ is known, $t$ t with $n - 1$ n − 1 degrees of freedom if not. 95% describes the method.
$X \sim Po (m)$ X ~ Po(m): independent events at a uniform average rate; mean = variance = $m$ m; independent Poisson variables add.
Binomial and Poisson tests are one-tailed. The critical region is as large as possible with $P (Type I error)$ P(Type I error) below the significance level.
$σ$ σ known: $z$ z-test; $σ$ σ unknown: $t$ t-test; paired data: one-sample test on the differences.
Good questions are unbiased, precise and structured; a reliable test is consistent (test-retest, parallel forms); a valid test measures the right thing (content, criterion-related).
$χ^{2}$ chi² goodness of fit: merge classes so expected frequencies exceed 5; df = categories − 1 − parameters estimated.
$S S_{res} = \sum (y - y^)^{2}$ SS_res = Σ (y − ŷ)² measures fit; $R^{2}$ R² is the proportion of variability explained; neither alone decides between models.

🏆

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You've worked through HL Statistics (AI HL) for IB Diploma Mathematics: Applications & Interpretation HL. 🎉

Mission 1A survey says students sleep 7.1 ± 0.2 hours. What does ± promise?
Mission 2Usually 3 complaints a week. This week, 8. Bad luck or a real problem?
Mission 3Two curves fit the same data. Which is better, and can you prove it?

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Specification points and sources for this lesson are listed at the bottom of the page.

HL Statistics (AI HL)

A survey says students sleep 7.1 ± 0.2 hours. What does ± promise?

Transforming and combining random variables

Packing a box of flour

Estimators and the central limit theorem

Confidence intervals for a mean

A tt-interval for mean sleep

Usually 3 complaints a week. This week, 8. Bad luck or a real problem?

The Poisson model

Two help desks

Testing a parameter; critical regions and errors

A binomial test for a proportion

Tests for a mean and for correlation

Two curves fit the same data. Which is better, and can you prove it?

Designing valid data collection

Reliability and validity

Residuals, SSres​SS_res and R2R²

SSres​SS_res for an exponential model

Big ideas

Mini-lesson complete!

📣 Smashed it? Share your score

A $t$ t-interval for mean sleep

Residuals, $S S_{res}$ SS_res and $R^{2}$ R²

$S S_{res}$ SS_res for an exponential model