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: 34, 29, 22, 13, 10, 42.

n = 6

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 50 and standard deviation 12. What proportion P of values is below 68?

μ = 50σ = 12x = 68

P =

3.ProbabilityMedium

For the distribution shown, find the standard deviation σ of X.

x: 1, 2, 3, 4, 5p(x): 0.1, 0.2, 0.05, 0.35, 0.3

σ =

4.ProbabilityHard

X has the distribution shown, and Y = 5X + 20. Find the standard deviation σ_Y of Y.

x: 0, 1, 2, 3p(x): 0.3, 0.2, 0.3, 0.2Y = 5X + 20

σ_Y =

5.Normal distributionMedium

X is normal with mean 250 g and standard deviation 4 g; Y is independent and normal with mean 500 g and standard deviation 9 g. Find the probability P that X + Y is more than 764.8 g. Round z to two decimals.

μ_X = 250, σ_X = 4μ_Y = 500, σ_Y = 9X + Y > 764.8 g

P =

6.Normal distributionHard

Runner A's lap time X and runner B's lap time Y are independent and normal: X has mean 52 s and standard deviation 4 s, Y has mean 48 s and standard deviation 3 s. Find the probability P that X is greater than Y. Round z to two decimals.

μ_X = 52 s, σ_X = 4 sμ_Y = 48 s, σ_Y = 3 s

P =

7.Confidence intervalsMedium

A sample of 16 items has mean 81.5 and standard deviation s = 1. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.

n = 16x̄ = 81.5s = 199%t(15) = 2.947

E =

8.Confidence intervalsHard

The population standard deviation is 9. 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 3?

σ = 9E = 395%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 64 students has a mean score of 78.15 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 75.

μ₀ = 75x̄ = 78.15s = 12n = 64

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.01 must detect a shift of the mean of 1.5 units with a power of 90%. The standard deviation is 10. Find the minimum sample size n.

α = 0.01power = 90%δ = 1.5σ = 10z_α = 2.326z_β = 1.282

n =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Find Pearson's correlation coefficient r.

n = 5

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 = 7.

n = 5x = 7

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 12.36

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 = 150/6 = 25

Squared deviations

Σ(x − x̄)² = 9² + 4² + (−3)² + (−12)² + (−15)² + 17² = 764

Standard deviation

s = √(764/5) = 12.36

Answer

s = 12.36

2.Descriptive statistics

Answer: P = 0.9332

Standardise

z = 1.5

Standard normal table

Φ(1.5) = 0.9332

P = 0.9332

Answer

P = 0.9332

3.Probability

Answer: σ = 1.359

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 = 1·0.1 + 2·0.2 + 3·0.05 + 4·0.35 + 5·0.3 = 3.55

Mean of the squares

E(X²) = 1·0.1 + 4·0.2 + 9·0.05 + 16·0.35 + 25·0.3 = 14.45

Variance

Var(X) = 14.45 − 3.55² = 1.848

σ = √1.848 = 1.359

Answer

σ = 1.359

4.Probability

Answer: σ_Y = 5.568

Spread of X

μ = 1.4, Var(X) = 1.24, σ_X = 1.114

Linear change

σ_Y = |5|·1.114 = 5.568

Answer

σ_Y = 5.568

The mean does move: E(Y) = 5·1.4 + 20 = 27.

5.Normal distribution

Answer: P = 0.06681

Sums and differences of normal variables: if X and Y are independent and normal, X + Y and X − Y are normal too. Means add or subtract: E(X ± Y) = μ_X ± μ_Y. Variances always add, also for a difference: Var(X ± Y) = σ_X² + σ_Y², so σ = √(σ_X² + σ_Y²). Standard deviations never add directly. The total of n independent items with the same distribution has mean n·μ and standard deviation σ·√n. Then standardise as usual: z = (x − mean)/σ and read Φ(z).

Distribution of the sum

μ = 750 g, σ = √(4² + 9²) = 9.849 g

Standardise

z = (764.8 − 750)/9.849 = 1.5

Upper tail

P = 1 − Φ(1.5) = 1 − 0.9332 = 0.06681

Answer

P = 0.06681

6.Normal distribution

Answer: P = 0.7881

Distribution of the difference

μ = 52 − 48 = 4 s, σ = 5 s

Standardise

z = (0 − 4)/5 = −0.8

Probability

P = 1 − Φ(−0.8) = 0.7881

Answer

P = 0.7881

7.Confidence intervals

Answer: E = 0.7368

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 = 15, t = 2.947

Margin of error

E = 2.947·1/√16 = 0.7368

Answer

E = 0.7368

8.Confidence intervals

Answer: n = 35

Solve for n

n = (1.96·9/3)² = 34.57

Round up

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

n = 35

Answer

n = 35

9.Hypothesis tests & confidence intervals

Answer: t = 2.1

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 = 12/√64 = 12/8 = 1.5

Test statistic

t = (78.15 − 75)/1.5 = 2.1

Answer

t = 2.1

t_{0.025; df = 63} = 1.998

|t| > 1.998, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: n = 579

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 + 1.282)·10/1.5)² = 578.6

Round up

n = 579

Answer

n = 579

11.Linear regression & correlation

Answer: r = 0.8802

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 = 120/5 = 24

ȳ = 323/5 = 64.6

Sums of squares and products

S_xx = Σ(x − x̄)² = 90

S_xy = Σ(x − x̄)(y − ȳ) = 234

S_yy = Σ(y − ȳ)² = 785.2

Correlation coefficient

r = 234/√(90·785.2) = 0.8802

Answer

r = 0.8802

12.Linear regression & correlation

Answer: ŷ = 28.7

Means

x̄ = 30/5 = 6

ȳ = 122/5 = 24.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 172

Regression line

b = 172/40 = 4.3

a = 24.4 − 4.3·6 = −1.4

ŷ = −1.4 + 4.3·x

Prediction

ŷ = −1.4 + 4.3·7 = 28.7

Answer

ŷ = 28.7

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