AP Statistics practice problems: cram pack

Formula sheets, the key method steps for every topic and 14 practice problems, with full solutions at the back.

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1Formulas and methods

Descriptive statistics

Mean: x̄ = Σx/n
Sample standard deviation: s = √(Σ(x − x̄)²/(n − 1))
Median: middle value of the sorted data
z-score: z = (x − μ)/σ
Normal distribution: 68% within ±1σ, 95% within ±2σ

Probability

Complement: P(not A) = 1 − P(A)
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent events: P(A ∩ B) = P(A)·P(B)
Conditional probability: P(A|B) = P(A ∩ B)/P(B)
Binomial: P(X = k) = C(n,k)·p^k(1 − p)^n−k

Normal distribution

z-score: z = (x − μ)/σ
Below x: P(X < x) = Φ(z)
Above x: P(X > x) = 1 − Φ(z)
Between: Φ(z₂) − Φ(z₁)
Percentile: x = μ + z·σ

Method

Convert the value to a z-score: z = (x − μ)/σ.
Look up Φ(z), the area to the left of z.
For the area to the right use 1 − Φ(z); for an interval subtract the two left areas.
For a percentile, work backwards: find z for the area, then x = μ + z·σ.

Linear regression & correlation

Sums of squares: S_xx = Σ(x − x̄)² = Σx² − (Σx)²/n, S_xy = Σxy − ΣxΣy/n
Slope: b = S_xy/S_xx
Intercept: a = ȳ − b·x̄
Correlation: r = S_xy/√(S_xxS_yy)
Residual: e = y − ŷ

Confidence intervals

σ known: x̄ ± z·σ/√n
σ unknown: x̄ ± t·s/√n, df = n − 1
Critical z: 90%: 1.645, 95%: 1.96, 99%: 2.576
Sample size: n = (z·σ/E)², rounded up

Method

Decide between z (population σ known) and t (only the sample s, with n − 1 degrees of freedom).
Find the critical value for the confidence level.
Margin of error: E = critical value × standard deviation/√n.
Interval: x̄ ± E.

Hypothesis tests & confidence intervals

z-test for a mean: z = (x̄ − μ₀)/(σ/√n)
t-test for a mean: t = (x̄ − μ₀)/(s/√n), df = n − 1
Two-sided at 5%: reject H₀ if |z| > 1.96
Confidence interval: x̄ ± z·σ/√n

Method

State H₀ and H₁ and decide whether the test is one- or two-sided.
Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
Compare with the critical value, or find the p-value.
Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.

Chi-square tests

Test statistic: χ² = Σ(O − E)²/E
Goodness of fit: E = n·p, df = k − 1 − (estimated parameters)
Independence: E = row total × column total / grand total, df = (r − 1)(c − 1)
Critical values (α = 0.05): df 1: 3.841, 2: 5.991, 3: 7.815, 4: 9.488
Decision: reject H₀ if χ² > χ²_crit

2Practice problems

1.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 22, 58, 18, 42, 60.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 170 and standard deviation 10. What proportion P of values is above 190?

μ = 170σ = 10x = 190

P =

3.ProbabilityMedium

A fair die is rolled 4 times. What is the probability P of getting at least one six?

n = 4

P =

4.ProbabilityHard

A test has 7 multiple-choice questions. A student who guesses gets each one right with probability 0.25, independently. What is the probability P of exactly 1 correct answers?

n = 7p = 0.25k = 1

P =

5.Normal distributionMedium

X is normal with mean 12 min and standard deviation 1.5 min; Y is independent and normal with mean 20 min and standard deviation 2 min. Find the probability P that X + Y is more than 34.5 min. Round z to two decimals.

μ_X = 12, σ_X = 1.5μ_Y = 20, σ_Y = 2X + Y > 34.5 min

P =

6.Normal distributionHard

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Drag the cut-off so the shaded bottom 25% is to the left of it. Find the cut-off x.

μ = 65 pointsσ = 10 points25%

x = points

7.Linear regression & correlationMedium

The table gives 5 paired observations of x = load in kN and y = extension in mm. Find Pearson's correlation coefficient r.

n = 5

r =

8.Linear regression & correlationHard

The table gives 5 paired observations of x = hours studied and y = exam score. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 69?

n = 5ŷ = 69

x =

9.Confidence intervalsMedium

A 90% confidence interval for a mean, based on 64 observations and a known σ, has a margin of error of 1.2. What is the margin of error E₂ if the same study uses 256 observations instead?

90%n₁ = 64E₁ = 1.2n₂ = 256

E₂ =

10.Confidence intervalsHard

The population standard deviation is 40. How large a sample n do you need so that a 95% confidence interval for the mean has a margin of error of at most 4?

σ = 40E = 495%

n =

11.Hypothesis tests & confidence intervalsMedium

A sample of 36 students has a mean score of 72.83 and a sample standard deviation of 5. Find the test statistic t for H₀: μ = 75.

μ₀ = 75x̄ = 72.83s = 5n = 36

t =

12.Hypothesis tests & confidence intervalsHard

A sample of 100 measurements has mean 121.4. The population standard deviation is 10. Find the upper limit U of the 99% confidence interval for the mean.

x̄ = 121.4σ = 10n = 10099%

U =

13.Chi-square testsMedium

The contingency table shows observed counts. For a chi-square test of independence, find the expected count E for the cell “staff, bike”.

row total = 93column total = 56N = 157

E =

14.Chi-square testsHard

The table shows observed counts. Test at α = 0.01 whether the two variables are independent (no continuity correction). Compute the test statistic χ², then compare it with the critical value from the table.

N = 1603 × 2α = 0.01

χ² =

3Solutions

1.Descriptive statistics

Answer: s = 19.6

Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.

Mean

x̄ = 200/5 = 40

Squared deviations

Σ(x − x̄)² = (−18)² + 18² + (−22)² + 2² + 20² = 1536

Standard deviation

s = √(1536/4) = 19.6

Answer

s = 19.6

2.Descriptive statistics

Answer: P = 0.02275

Standardise

z = 2

Standard normal table

Φ(2) = 0.9772

P = 1 − 0.9772 = 0.02275

Answer

P = 0.02275

3.Probability

Answer: P = 0.5177

Probability: with equally likely outcomes, P(A) = favourable/total. For independent events multiply: P(A and B) = P(A)·P(B). The complement often helps: P(at least one) = 1 − P(none). Binomial: in n independent trials with success probability p, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ.

Complement

P(no six) = (5/6)⁴ = 0.4823

P = 1 − 0.4823 = 0.5177

Answer

P = 0.5177

4.Probability

Answer: P = 0.3115

Binomial formula

P(X = 1) = 7·0.25¹·0.75⁶ = 7·0.25·0.178 = 0.3115

Answer

P = 0.3115

5.Normal distribution

Answer: P = 0.1587

Sums and differences of normal variables: if X and Y are independent and normal, X + Y and X − Y are normal too. Means add or subtract: E(X ± Y) = μ_X ± μ_Y. Variances always add, also for a difference: Var(X ± Y) = σ_X² + σ_Y², so σ = √(σ_X² + σ_Y²). Standard deviations never add directly. The total of n independent items with the same distribution has mean n·μ and standard deviation σ·√n. Then standardise as usual: z = (x − mean)/σ and read Φ(z).

Distribution of the sum

μ = 32 min, σ = √(1.5² + 2²) = 2.5 min

Standardise

z = (34.5 − 32)/2.5 = 1

Upper tail

P = 1 − Φ(1) = 1 − 0.8413 = 0.1587

Answer

P = 0.1587

6.Normal distribution

Answer: x = 58.3 points

From an area back to z: Φ(z) is the area to the left of z. For a left-tail area A, find z with Φ(z) = A in the table. A right-tail area A means Φ(z) = 1 − A. A central area A between −z and z leaves (1 − A)/2 in each tail, so Φ(z) = (1 + A)/2. For a value x, use x = μ + zσ.

Area to the left

Φ(z) = 0.25

z from the table

z ≈ −0.67

Back to the original units

x = 65 − 0.67·10 = 58.3 points

Answer

x = 58.3 points

7.Linear regression & correlation

Answer: r = 0.9957

Least-squares regression: with x̄ and ȳ the means, Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)² and Sxy = Σ(x − x̄)(y − ȳ), the line ŷ = a + b·x has slope b = Sxy/Sxx and intercept a = ȳ − b·x̄, so it always passes through (x̄, ȳ). Pearson's correlation coefficient r = Sxy/√(Sxx·Syy) lies between −1 and 1; r² is the fraction of the variation in y explained by the line.

Means

x̄ = 30/5 = 6

ȳ = 71/5 = 14.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 94

S_yy = Σ(y − ȳ)² = 222.8

Correlation coefficient

r = 94/√(40·222.8) = 0.9957

Answer

r = 0.9957

8.Linear regression & correlation

Answer: x = 6.091

Means

x̄ = 25/5 = 5

ȳ = 324/5 = 64.8

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 154

Regression line

b = 154/40 = 3.85

a = 64.8 − 3.85·5 = 45.55

ŷ = 45.55 + 3.85·x

Solve for x

x = (69 − 45.55)/3.85 = 6.091

Answer

x = 6.091

9.Confidence intervals

Answer: E₂ = 0.6

Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.

Ratio of margins

E₂/E₁ = (1.645/1.645)·√(64/256) = 0.5

New margin

E₂ = 1.2·0.5 = 0.6

Answer

E₂ = 0.6

10.Confidence intervals

Answer: n = 385

Solve for n

n = (1.96·40/4)² = 384.2

Round up

A sample size is a whole number, and rounding down would make the margin too large.

n = 385

Answer

n = 385

11.Hypothesis tests & confidence intervals

Answer: t = −2.604

Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).

Standard error

SE = 5/√36 = 5/6 = 0.8333

Test statistic

t = (72.83 − 75)/0.8333 = −2.604

Answer

t = −2.604

t_{0.025; df = 35} = 2.03

|t| > 2.03, so H₀ is rejected at the 5% level (two-sided).

12.Hypothesis tests & confidence intervals

Answer: U = 124

Margin of error

E = 2.576·10/√100 = 2.576

Interval limit

U = 121.4 + 2.576 = 124

Answer

U = 124

13.Chi-square tests

Answer: E = 33.17

Chi-square tests: compare observed counts O with the counts E expected under H₀ using χ² = Σ(O − E)²/E. Goodness of fit: E = n·p for each category, df = k − 1 (minus one more for each parameter estimated from the data). Independence in an r × c table: E = row total × column total / grand total, df = (r − 1)(c − 1). Reject H₀ when χ² exceeds the critical value for the chosen α.

Expected count

E = 93·56/157 = 33.17

Answer

E = 33.17

14.Chi-square tests

Answer: χ² = 27.78

Expected counts

plant A: 36, 24

plant B: 36, 24

plant C: 24, 16

Chi-square statistic

χ² = (41 − 36)²/36 + (19 − 24)²/24 + (45 − 36)²/36 + (15 − 24)²/24 + (10 − 24)²/24 + (30 − 16)²/16

χ² = 27.78

Decision

df = 2, χ²_crit = 9.210

27.78 > 9.210

Answer

χ² = 27.78

χ² is above the critical value, so H₀ is rejected: the variables appear to be related.

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