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: 20, 28, 31, 31, 45, 19, 18.

n = 7

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 70 and standard deviation 12. What proportion P of values is above 74.8?

μ = 70σ = 12x = 74.8

P =

3.ProbabilityMedium

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

x: 0, 1, 2, 3, 4p(x): 0.05, 0.3, 0.2, 0.2, 0.25

Var(X) =

4.ProbabilityHard

X has the distribution shown, and Y = −2X + 4. Find the standard deviation σ_Y of Y.

x: 10, 20, 30, 40p(x): 0.35, 0.3, 0.25, 0.1Y = −2X + 4

σ_Y =

5.Normal distributionMedium

X is normal with mean 60 s and standard deviation 5 s; Y is independent and normal with mean 45 s and standard deviation 4 s. Find the probability P that X + Y is more than 108.2 s. Round z to two decimals.

μ_X = 60, σ_X = 5μ_Y = 45, σ_Y = 4X + Y > 108.2 s

P =

6.Normal distributionHard

The mass X of a product from line 1 and Y from line 2 are independent and normal: X has mean 500 g and standard deviation 8 g, Y has mean 490 g and standard deviation 6 g. Find the probability P that X is greater than Y. Round z to two decimals.

μ_X = 500 g, σ_X = 8 gμ_Y = 490 g, σ_Y = 6 g

P =

7.Confidence intervalsMedium

A 99% confidence interval for a mean, based on 36 observations and a known σ, has a margin of error of 7.6. What is the margin of error E₂ of a 95% interval based on 324 observations?

99%n₁ = 36E₁ = 7.695%n₂ = 324

E₂ =

8.Confidence intervalsHard

The population standard deviation is 40. 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 0.5?

σ = 40E = 0.599%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 16 students has a mean score of 114.8 and a sample standard deviation of 8. Find the test statistic t for H₀: μ = 120.

μ₀ = 120x̄ = 114.8s = 8n = 16

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test of H₀: μ = 500 against H₁: μ > 500 is done at α = 0.05 with a sample of 64; the standard deviation is known to be 4. In reality the mean is 501.5. Find the probability of a Type II error β in percent.

μ₀ = 500μ = 501.5σ = 4n = 64α = 0.05

β = %

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = car age in years and y = price in k€. Find Pearson's correlation coefficient r.

n = 5

r =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 4.

n = 6x = 4

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 9.572

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 = 192/7 = 27.43

Squared deviations

Σ(x − x̄)² = (−7.429)² + 0.5714² + 3.571² + 3.571² + 17.57² + (−8.429)² + (−9.429)² = 549.7

Standard deviation

s = √(549.7/6) = 9.572

Answer

s = 9.572

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: Var(X) = 1.61

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 = 0·0.05 + 1·0.3 + 2·0.2 + 3·0.2 + 4·0.25 = 2.3

Mean of the squares

E(X²) = 0·0.05 + 1·0.3 + 4·0.2 + 9·0.2 + 16·0.25 = 6.9

Variance

Var(X) = 6.9 − 2.3² = 1.61

Answer

Var(X) = 1.61

4.Probability

Answer: σ_Y = 19.9

Spread of X

μ = 21, Var(X) = 99, σ_X = 9.95

Linear change

σ_Y = |−2|·9.95 = 19.9

Answer

σ_Y = 19.9

The mean does move: E(Y) = −2·21 + 4 = −38.

5.Normal distribution

Answer: P = 0.3085

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 = 105 s, σ = √(5² + 4²) = 6.403 s

Standardise

z = (108.2 − 105)/6.403 = 0.5

Upper tail

P = 1 − Φ(0.5) = 1 − 0.6915 = 0.3085

Answer

P = 0.3085

6.Normal distribution

Answer: P = 0.8413

Distribution of the difference

μ = 500 − 490 = 10 g, σ = 10 g

Standardise

z = (0 − 10)/10 = −1

Probability

P = 1 − Φ(−1) = 0.8413

Answer

P = 0.8413

7.Confidence intervals

Answer: E₂ = 1.928

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 = (1.96/2.576)·√(36/324) = 0.2536

New margin

E₂ = 7.6·0.2536 = 1.928

Answer

E₂ = 1.928

8.Confidence intervals

Answer: n = 42469

Solve for n

n = (2.576·40/0.5)² = 42469

Round up

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

n = 42469

Answer

n = 42469

9.Hypothesis tests & confidence intervals

Answer: t = −2.6

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 = 8/√16 = 8/4 = 2

Test statistic

t = (114.8 − 120)/2 = −2.6

Answer

t = −2.6

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: β = 8.771 %

Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).

Critical sample mean

SE = 4/√64 = 0.5

x̄_c = 500 + 1.645·0.5 = 500.8

z under the true mean

z = (500.8 − 501.5)/0.5 = −1.355

Type II error

power = P(Z > −1.355) = 91.23%

β = 100% − 91.23% = 8.771%

Answer

β = 8.771%

11.Linear regression & correlation

Answer: r = −0.9117

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 = 15/5 = 3

ȳ = 112/5 = 22.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = −25

S_yy = Σ(y − ȳ)² = 75.2

Correlation coefficient

r = −25/√(10·75.2) = −0.9117

Answer

r = −0.9117

12.Linear regression & correlation

Answer: ŷ = 47.01

Means

x̄ = 36/6 = 6

ȳ = 333/6 = 55.5

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 297

Regression line

b = 297/70 = 4.243

a = 55.5 − 4.243·6 = 30.04

ŷ = 30.04 + 4.243·x

Prediction

ŷ = 30.04 + 4.243·4 = 47.01

Answer

ŷ = 47.01

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