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: 41, 59, 46, 23, 34, 26, 37.

n = 7

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 70 and standard deviation 8. What proportion P of values is above 78?

μ = 70σ = 8x = 78

P =

3.ProbabilityMedium

In how many ways N can a committee of 4 people be chosen from a group of 8? The order does not matter.

n = 8k = 4

N =

4.ProbabilityHard

A test has 5 multiple-choice questions. A student who guesses gets each one right with probability 0.4, independently. What is the probability P of exactly 2 correct answers?

n = 5p = 0.4k = 2

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.9715. Find z to two decimals.

A = 0.9715

z =

6.Normal distributionHard

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. Find the value x below which 5% of them lie (the 5th percentile).

μ = 500σ = 40P = 5%

x = h

7.Confidence intervalsMedium

A sample of 20 items has mean 30.5 and standard deviation s = 17.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.

n = 20x̄ = 30.5s = 17.599%t(19) = 2.861

E =

8.Confidence intervalsHard

The population standard deviation is 29. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 1?

σ = 29E = 199%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 16 students has a mean score of 493.8 and a sample standard deviation of 8. Find the test statistic t for H₀: μ = 500.

μ₀ = 500x̄ = 493.8s = 8n = 16

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.01 must detect a shift of the mean of 3 units with a power of 80%. The standard deviation is 20. Find the minimum sample size n.

α = 0.01power = 80%δ = 3σ = 20z_α = 2.326z_β = 0.842

n =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Find the coefficient of determination r² of the least-squares line.

n = 5

r² =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = hours studied and y = exam score. Use the least-squares line ŷ = a + b·x. A new observation (7, 78) is made. Find its residual e = y − ŷ.

n = 6(7, 78)

e =

3Solutions

1.Descriptive statistics

Answer: s = 12.25

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 = 266/7 = 38

Squared deviations

Σ(x − x̄)² = 3² + 21² + 8² + (−15)² + (−4)² + (−12)² + (−1)² = 900

Standard deviation

s = √(900/6) = 12.25

Answer

s = 12.25

2.Descriptive statistics

Answer: P = 0.1587

Standardise

z = 1

Standard normal table

Φ(1) = 0.8413

P = 1 − 0.8413 = 0.1587

Answer

P = 0.1587

3.Probability

Answer: N = 70

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, 4) = (8·7·6·5)/(4·3·2·1) = 70

Answer

N = 70

4.Probability

Answer: P = 0.3456

Binomial formula

P(X = 2) = 10·0.4²·0.6³ = 10·0.16·0.216 = 0.3456

Answer

P = 0.3456

5.Normal distribution

Answer: z = 2.19

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 + 0.9715)/2 = 0.9857

Read the z-table backwards

Find 0.9857 inside the table and read off z.

z = 2.19

Answer

z = 2.19

6.Normal distribution

Answer: x = 434.2 h

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) = 0.05 → z = −1.645

Back to x

x = 500 + −1.645·40 = 434.2 h

Answer

x = 434.2 h

7.Confidence intervals

Answer: E = 11.2

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 = 2.861

Margin of error

E = 2.861·17.5/√20 = 11.2

Answer

E = 11.2

8.Confidence intervals

Answer: n = 5581

Solve for n

n = (2.576·29/1)² = 5581

Round up

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

n = 5581

Answer

n = 5581

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 = 8/√16 = 8/4 = 2

Test statistic

t = (493.8 − 500)/2 = −3.1

Answer

t = −3.1

t_{0.025; df = 15} = 2.131

|t| > 2.131, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: n = 447

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)·20/3)² = 446.1

Round up

n = 447

Answer

n = 447

11.Linear regression & correlation

Answer: r² = 0.8329

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 = 25/5 = 5

ȳ = 206/5 = 41.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 68

S_yy = Σ(y − ȳ)² = 138.8

Coefficient of determination

r² = 68²/(40·138.8) = 0.8329

Answer

r² = 0.8329

About 83.29% of the variation in y is explained by the line.

12.Linear regression & correlation

Answer: e = 2.667

Means

x̄ = 42/6 = 7

ȳ = 452/6 = 75.33

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 264

Regression line

b = 264/70 = 3.771

a = 75.33 − 3.771·7 = 48.93

ŷ = 48.93 + 3.771·x

Predicted value

ŷ = 48.93 + 3.771·7 = 75.33

Residual

e = 78 − 75.33 = 2.667

Answer

e = 2.667

The point lies above the line.

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