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: 55, 59, 56, 26, 24, 48, 14.

n = 7

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 170 and standard deviation 12. Find the z-score z of a score of 155.6.

μ = 170σ = 12x = 155.6

z =

3.ProbabilityMedium

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

x: 10, 20, 30, 40p(x): 0.3, 0.25, 0.35, 0.1

Var(X) =

4.ProbabilityHard

A test has 6 multiple-choice questions. A student who guesses gets each one right with probability 0.2, independently. What is the probability P of exactly 2 correct answers?

n = 6p = 0.2k = 2

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.4313. Find z to two decimals.

A = 0.4313

z =

6.Normal distributionHard

The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. Find the value x below which 95% of them lie (the 95th percentile).

μ = 100σ = 15P = 95%

x = points

7.Confidence intervalsMedium

A sample of 25 items has mean 60.5 and standard deviation s = 16. The population is roughly normal. Find the margin of error E of a 95% confidence interval for the mean.

n = 25x̄ = 60.5s = 1695%t(24) = 2.064

E =

8.Confidence intervalsHard

The population standard deviation is 23. 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 = 23E = 495%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 64 students has a mean score of 200.3 and a sample standard deviation of 4. Find the test statistic t for H₀: μ = 200.

μ₀ = 200x̄ = 200.3s = 4n = 64

t =

10.Hypothesis tests & confidence intervalsHard

A sample of 36 measurements has mean 201.1. The population standard deviation is 4. Find the lower limit L of the 90% confidence interval for the mean.

x̄ = 201.1σ = 4n = 3690%

L =

11.Linear regression & correlationMedium

The table gives 5 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 = 5Σx = 35Σy = 364Σx² = 285Σy² = 27606Σxy = 2756

r =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 11.

n = 6x = 11

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 18.41

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 = 282/7 = 40.29

Squared deviations

Σ(x − x̄)² = 14.71² + 18.71² + 15.71² + (−14.29)² + (−16.29)² + 7.714² + (−26.29)² = 2033

Standard deviation

s = √(2033/6) = 18.41

Answer

s = 18.41

2.Descriptive statistics

Answer: z = −1.2

Standardise

z = (155.6 − 170)/12 = −1.2

Answer

z = −1.2

3.Probability

Answer: Var(X) = 98.75

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 = 10·0.3 + 20·0.25 + 30·0.35 + 40·0.1 = 22.5

Mean of the squares

E(X²) = 100·0.3 + 400·0.25 + 900·0.35 + 1600·0.1 = 605

Variance

Var(X) = 605 − 22.5² = 98.75

Answer

Var(X) = 98.75

4.Probability

Answer: P = 0.2458

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.

Binomial formula

P(X = 2) = 15·0.2²·0.8⁴ = 15·0.04·0.4096 = 0.2458

Answer

P = 0.2458

5.Normal distribution

Answer: z = 0.57

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 of z

Φ(z) = (1 + 0.4313)/2 = 0.7157

Read the z-table backwards

Find 0.7157 inside the table and read off z.

z = 0.57

Answer

z = 0.57

6.Normal distribution

Answer: x = 124.7 points

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 for the percentile

Φ(z) = 0.95 → z = 1.645

Back to x

x = 100 + 1.645·15 = 124.67 points

Answer

x = 124.67 points

7.Confidence intervals

Answer: E = 6.605

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.

Critical value

df = n − 1 = 24, t = 2.064

Margin of error

E = 2.064·16/√25 = 6.605

Answer

E = 6.605

8.Confidence intervals

Answer: n = 128

Solve for n

n = (1.96·23/4)² = 127

Round up

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

n = 128

Answer

n = 128

9.Hypothesis tests & confidence intervals

Answer: t = 0.6

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 = 4/√64 = 4/8 = 0.5

Test statistic

t = (200.3 − 200)/0.5 = 0.6

Answer

t = 0.6

t_{0.025; df = 63} = 1.998

|t| < 1.998, so H₀ is not rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: L = 200

Margin of error

E = 1.645·4/√36 = 1.097

Interval limit

L = 201.1 − 1.097 = 200

Answer

L = 200

11.Linear regression & correlation

Answer: r = 0.9886

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 = 2756 − 35·364/5 = 208

S_xx = 285 − 35²/5 = 40

S_yy = 27606 − 364²/5 = 1107

Correlation coefficient

r = 208/√(40·1107) = 0.9886

Answer

r = 0.9886

12.Linear regression & correlation

Answer: ŷ = 18.94

Means

x̄ = 42/6 = 7

ȳ = 78/6 = 13

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 104

Regression line

b = 104/70 = 1.486

a = 13 − 1.486·7 = 2.6

ŷ = 2.6 + 1.486·x

Prediction

ŷ = 2.6 + 1.486·11 = 18.94

Answer

ŷ = 18.94

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