Introductory Statistics: cram pack

Formula sheets, the key method steps for every topic and 12 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.

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.

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

2Practice problems

1.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 42, 52, 32, 35, 49, 30.

n = 6

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 70 and standard deviation 12. Find the z-score z of a score of 82.

μ = 70σ = 12x = 82

z =

3.ProbabilityMedium

In how many ways N can a committee of 2 people be chosen from a group of 8? The order does not matter.

n = 8k = 2

N =

4.ProbabilityHard

A test has 5 multiple-choice questions. A student who guesses gets each one right with probability 0.2, independently. What is the probability P of exactly 1 correct answers?

n = 5p = 0.2k = 1

P =

5.Normal distributionMedium

The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. What fraction lies between 174.85 and 188.5 cm? Give P as a decimal.

μ = 178σ = 7x₁ = 174.85x₂ = 188.5

P =

6.Normal distributionHard

The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. Drag the cut-off so the shaded bottom 95% is to the left of it. Find the cut-off x.

μ = 100 pointsσ = 15 points95%

x = points

7.Confidence intervalsMedium

A 95% confidence interval for a mean, based on 36 observations and a known σ, has a margin of error of 7.8. What is the margin of error E₂ of a 90% interval based on 144 observations?

95%n₁ = 36E₁ = 7.890%n₂ = 144

E₂ =

8.Confidence intervalsHard

Two fertilisers are tested. Plot group 1: 7 plots, mean yield 67 kg, standard deviation 4 kg. Plot group 2: 7 plots, mean 69.5 kg, standard deviation 3 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the lower limit of the 99% confidence interval for μ₁ − μ₂.

n₁ = 7x̄₁ = 67s₁ = 4n₂ = 7x̄₂ = 69.5s₂ = 399%t(12) = 3.055

lower limit = kg

9.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 111.2 and a sample standard deviation of 20. Find the test statistic t for H₀: μ = 120.

μ₀ = 120x̄ = 111.2s = 20n = 25

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.05 must detect a shift of the mean of 1.5 units with a power of 80%. The standard deviation is 4. Find the minimum sample size n.

α = 0.05power = 80%δ = 1.5σ = 4z_α = 1.645z_β = 0.842

n =

11.Linear regression & correlationMedium

The table gives 6 paired observations of x = advertising spend in k€ and y = sales in k€. Their sums are given. Find Pearson's correlation coefficient r from these sums.

n = 6Σx = 33Σy = 305Σx² = 199Σy² = 15761Σxy = 1739

r =

12.Linear regression & correlationHard

The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. A new observation (18, 50) is made. Find its residual e = y − ŷ.

n = 5(18, 50)

e =

3Solutions

1.Descriptive statistics

Answer: s = 9.143

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 = 240/6 = 40

Squared deviations

Σ(x − x̄)² = 2² + 12² + (−8)² + (−5)² + 9² + (−10)² = 418

Standard deviation

s = √(418/5) = 9.143

Answer

s = 9.143

2.Descriptive statistics

Answer: z = 1

Standardise

z = (82 − 70)/12 = 1

Answer

z = 1

3.Probability

Answer: N = 28

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.

Combinations

C(8, 2) = (8·7)/(2·1) = 28

Answer

N = 28

4.Probability

Answer: P = 0.4096

Binomial formula

P(X = 1) = 5·0.2¹·0.8⁴ = 5·0.2·0.4096 = 0.4096

Answer

P = 0.4096

5.Normal distribution

Answer: P = 0.6068

Normal distribution: convert a value x to a z-score, z = (x − μ)/σ, which counts standard deviations from the mean. A z-table (or Φ(z)) gives the area to the left of z, which is P(X < x). The area to the right is 1 − Φ(z), and the area between two values is the difference of their two areas.

z-scores

z₁ = (174.85 − 178)/7 = −0.45

z₂ = (188.5 − 178)/7 = 1.5

Area between

P = Φ(1.5) − Φ(−0.45) = 0.9332 − 0.3264 = 0.6068

Answer

P = 0.6068

6.Normal distribution

Answer: x = 124.6 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) = 0.95

z from the table

z ≈ 1.64

Back to the original units

x = 100 + 1.64·15 = 124.6 points

Answer

x = 124.6 points

7.Confidence intervals

Answer: E₂ = 3.273

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 = (1.645/1.96)·√(36/144) = 0.4196

New margin

E₂ = 7.8·0.4196 = 3.273

Answer

E₂ = 3.273

8.Confidence intervals

Answer: lower limit = −8.273 kg

Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.

Pooled standard deviation

s_p² = (6·4² + 6·3²)/12 = 12.5

s_p = 3.536

Margin of error

E = 3.055·3.536·√(1/7 + 1/7) = 5.773

Interval

−2.5 ± 5.773 → [−8.273, 3.273]

Answer

lower limit = −8.273 kg

9.Hypothesis tests & confidence intervals

Answer: t = −2.2

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 = 20/√25 = 20/5 = 4

Test statistic

t = (111.2 − 120)/4 = −2.2

Answer

t = −2.2

t_{0.025; df = 24} = 2.064

|t| > 2.064, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: n = 44

Sample size for a test: to detect a true shift δ with power 1 − β in a one-sided z test at level α, the standard error must satisfy δ = (z_α + z_β)·σ/√n, so n = ((z_α + z_β)·σ/δ)², rounded up.

Solve for n

n = ((1.645 + 0.842)·4/1.5)² = 43.98

Round up

n = 44

Answer

n = 44

11.Linear regression & correlation

Answer: r = 0.9173

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.

Centred sums

S_xy = 1739 − 33·305/6 = 61.5

S_xx = 199 − 33²/6 = 17.5

S_yy = 15761 − 305²/6 = 256.8

Correlation coefficient

r = 61.5/√(17.5·256.8) = 0.9173

Answer

r = 0.9173

12.Linear regression & correlation

Answer: e = 2

Means

x̄ = 105/5 = 21

ȳ = 291/5 = 58.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 136

Regression line

b = 136/40 = 3.4

a = 58.2 − 3.4·21 = −13.2

ŷ = −13.2 + 3.4·x

Predicted value

ŷ = −13.2 + 3.4·18 = 48

Residual

e = 50 − 48 = 2

Answer

e = 2

The point lies above the line.

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