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: 27, 37, 54, 52, 26.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 60 and standard deviation 8. What proportion P of values is above 63.2?

μ = 60σ = 8x = 63.2

P =

3.ProbabilityMedium

In how many ways N can a committee of 5 people be chosen from a group of 11? The order does not matter.

n = 11k = 5

N =

4.ProbabilityHard

10 components are tested, and each one fails with probability 0.3, independently. What is the probability P that at most one fails?

n = 10p = 0.3

P =

5.Normal distributionMedium

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. What fraction lies between 452 and 510 h? Give P as a decimal.

μ = 500σ = 40x₁ = 452x₂ = 510

P =

6.Normal distributionHard

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. Drag the cut-off so the shaded top 10% is to the right of it. Find the cut-off x.

μ = 500 hσ = 40 h10%

x = h

7.Confidence intervalsMedium

A sample of 20 items has mean 111 and standard deviation s = 14. The population is roughly normal. Find the upper limit of the 90% confidence interval for the mean.

n = 20x̄ = 111s = 1490%t(19) = 1.729

upper limit =

8.Confidence intervalsHard

Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 99% confidence interval for the proportion has a margin of error of at most 0.05?

p = 0.5E = 0.0599%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 49 students has a mean score of 194.7 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 200.

μ₀ = 200x̄ = 194.7s = 12n = 49

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.01 must detect a shift of the mean of 2.5 units with a power of 80%. The standard deviation is 14. Find the minimum sample size n.

α = 0.01power = 80%δ = 2.5σ = 14z_α = 2.326z_β = 0.842

n =

11.Linear regression & correlationMedium

The table gives 5 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 = 5Σx = 15Σy = 104Σx² = 55Σy² = 2240Σxy = 285

r =

12.Linear regression & correlationHard

The table gives 5 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 = 3.

n = 5x = 3

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 13.33

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 = 196/5 = 39.2

Squared deviations

Σ(x − x̄)² = (−12.2)² + (−2.2)² + 14.8² + 12.8² + (−13.2)² = 710.8

Standard deviation

s = √(710.8/4) = 13.33

Answer

s = 13.33

2.Descriptive statistics

Answer: P = 0.3446

Standardise

z = 0.4

Standard normal table

Φ(0.4) = 0.6554

P = 1 − 0.6554 = 0.3446

Answer

P = 0.3446

3.Probability

Answer: N = 462

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(11, 5) = (11·10·9·8·7)/(5·4·3·2·1) = 462

Answer

N = 462

4.Probability

Answer: P = 0.1493

Zero and one failure

P(X = 0) = 0.7¹⁰ = 0.02825

P(X = 1) = 10·0.3·0.7⁹ = 0.1211

Add them

P = 0.02825 + 0.1211 = 0.1493

Answer

P = 0.1493

5.Normal distribution

Answer: P = 0.4836

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 = (452 − 500)/40 = −1.2

z₂ = (510 − 500)/40 = 0.25

Area between

P = Φ(0.25) − Φ(−1.2) = 0.5987 − 0.1151 = 0.4836

Answer

P = 0.4836

6.Normal distribution

Answer: x = 551 h

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) = 0.9

z from the table

z ≈ 1.28

Back to the original units

x = 500 + 1.28·40 = 551 h

Answer

x = 551 h

7.Confidence intervals

Answer: upper limit = 116.4

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 = 19, t = 1.729

Margin of error

E = 1.729·14/√20 = 5.413

Interval

111 ± 5.413 → [105.6, 116.4]

Answer

upper limit = 116.4

8.Confidence intervals

Answer: n = 664

Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.

Solve for n

n = 2.576²·0.5·0.5/0.05² = 663.6

Round up

n = 664

Answer

n = 664

9.Hypothesis tests & confidence intervals

Answer: t = −3.092

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 = 12/√49 = 12/7 = 1.714

Test statistic

t = (194.7 − 200)/1.714 = −3.092

Answer

t = −3.092

t_{0.025; df = 48} = 2.011

|t| > 2.011, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: n = 315

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 = ((2.326 + 0.842)·14/2.5)² = 314.7

Round up

n = 315

Answer

n = 315

11.Linear regression & correlation

Answer: r = −0.9743

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 = 285 − 15·104/5 = −27

S_xx = 55 − 15²/5 = 10

S_yy = 2240 − 104²/5 = 76.8

Correlation coefficient

r = −27/√(10·76.8) = −0.9743

Answer

r = −0.9743

12.Linear regression & correlation

Answer: ŷ = 11.55

Means

x̄ = 30/5 = 6

ȳ = 99/5 = 19.8

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 110

Regression line

b = 110/40 = 2.75

a = 19.8 − 2.75·6 = 3.3

ŷ = 3.3 + 2.75·x

Prediction

ŷ = 3.3 + 2.75·3 = 11.55

Answer

ŷ = 11.55

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