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: 13, 37, 22, 55, 44.

n = 5

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 500 and standard deviation 60. Find the z-score z of a score of 428.

μ = 500σ = 60x = 428

z =

3.ProbabilityMedium

For the distribution shown, find the variance Var(X) of X.

x: 0, 1, 2, 3, 4p(x): 0.2, 0.3, 0.1, 0.2, 0.2

Var(X) =

4.ProbabilityHard

A game costs €5 to play. You win €25 with probability 0.3 and €100 with probability 0.02; otherwise you win nothing. The stake is not returned. Find the expected net gain E(G) per game.

stake = €5€25: p = 0.3€100: p = 0.02

E(G) = €

5.Normal distributionMedium

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. What fraction lies between 440 and 508 h? Give P as a decimal.

μ = 500σ = 40x₁ = 440x₂ = 508

P =

6.Normal distributionHard

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Drag the cut-off so the shaded top 85% is to the right of it. Find the cut-off x.

μ = 250 gσ = 5 g85%

x = g

7.Confidence intervalsMedium

A 90% confidence interval for a mean, based on 100 observations and a known σ, has a margin of error of 11.4. What is the margin of error E₂ of a 99% interval based on 900 observations?

90%n₁ = 100E₁ = 11.499%n₂ = 900

E₂ =

8.Confidence intervalsHard

The population standard deviation is 29. How large a sample n do you need so that a 90% confidence interval for the mean has a margin of error of at most 1?

σ = 29E = 190%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 62.6 and a sample standard deviation of 20. Find the test statistic t for H₀: μ = 75.

μ₀ = 75x̄ = 62.6s = 20n = 25

t =

10.Hypothesis tests & confidence intervalsHard

How large must a sample be so that the 95% confidence interval for a mean has a margin of error of at most 0.5? The standard deviation is 20. Find the smallest sample size n.

σ = 20E = 0.595%

n =

11.Linear regression & correlationMedium

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

n = 6Σx = 21Σy = 107Σx² = 91Σy² = 2135Σxy = 312

r =

12.Linear regression & correlationHard

The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 48?

n = 5ŷ = 48

x =

3Solutions

1.Descriptive statistics

Answer: s = 16.84

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 = 171/5 = 34.2

Squared deviations

Σ(x − x̄)² = (−21.2)² + 2.8² + (−12.2)² + 20.8² + 9.8² = 1135

Standard deviation

s = √(1135/4) = 16.84

Answer

s = 16.84

2.Descriptive statistics

Answer: z = −1.2

Standardise

z = (428 − 500)/60 = −1.2

Answer

z = −1.2

3.Probability

Answer: Var(X) = 2.09

Discrete random variables: a probability distribution lists every value x a variable can take with its probability p(x); the probabilities are between 0 and 1 and add up to 1. The expected value (mean) is E(X) = Σx·p(x), the long-run average. The variance is Var(X) = Σ(x − μ)²·p(x) = E(X²) − μ², and the standard deviation σ = √Var(X). For a linear change Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X) and σ_Y = |a|·σ_X; adding a constant does not change the spread.

Mean

μ = 0·0.2 + 1·0.3 + 2·0.1 + 3·0.2 + 4·0.2 = 1.9

Mean of the squares

E(X²) = 0·0.2 + 1·0.3 + 4·0.1 + 9·0.2 + 16·0.2 = 5.7

Variance

Var(X) = 5.7 − 1.9² = 2.09

Answer

Var(X) = 2.09

4.Probability

Answer: E(G) = 4.5 €

Net outcomes

G: 20 (p = 0.3), 95 (p = 0.02), −5 (p = 0.68)

Expected value

E(G) = 20·0.3 + 95·0.02 − 5·0.68 = 4.5 €

Answer

E(G) = 4.5 €

A positive value means you win money on average.

5.Normal distribution

Answer: P = 0.5125

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 = (440 − 500)/40 = −1.5

z₂ = (508 − 500)/40 = 0.2

Area between

P = Φ(0.2) − Φ(−1.5) = 0.5793 − 0.06681 = 0.5125

Answer

P = 0.5125

6.Normal distribution

Answer: x = 244.8 g

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) = 0.15

z from the table

z ≈ −1.04

Back to the original units

x = 250 − 1.04·5 = 244.8 g

Answer

x = 244.8 g

7.Confidence intervals

Answer: E₂ = 5.951

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 = (2.576/1.645)·√(100/900) = 0.522

New margin

E₂ = 11.4·0.522 = 5.951

Answer

E₂ = 5.951

8.Confidence intervals

Answer: n = 2276

Solve for n

n = (1.645·29/1)² = 2276

Round up

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

n = 2276

Answer

n = 2276

9.Hypothesis tests & confidence intervals

Answer: t = −3.1

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 = (62.6 − 75)/4 = −3.1

Answer

t = −3.1

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 = 6147

Solve for n

n = (1.96·20/0.5)² = 6147

Round up

n = 6147

Answer

n = 6147

11.Linear regression & correlation

Answer: r = −0.992

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 = 312 − 21·107/6 = −62.5

S_xx = 91 − 21²/6 = 17.5

S_yy = 2135 − 107²/6 = 226.8

Correlation coefficient

r = −62.5/√(17.5·226.8) = −0.992

Answer

r = −0.992

12.Linear regression & correlation

Answer: x = 3.426

Means

x̄ = 15/5 = 3

ȳ = 227/5 = 45.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = 61

Regression line

b = 61/10 = 6.1

a = 45.4 − 6.1·3 = 27.1

ŷ = 27.1 + 6.1·x

Solve for x

x = (48 − 27.1)/6.1 = 3.426

Answer

x = 3.426

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