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: 19, 36, 33, 28, 47.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 500 and standard deviation 60. What proportion P of values is below 470?

μ = 500σ = 60x = 470

P =

3.ProbabilityMedium

A bag holds 3 red and 9 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?

3 / 9

P =

4.ProbabilityHard

A test has 11 multiple-choice questions. A student who guesses gets each one right with probability 0.2, independently. What is the probability P of exactly 5 correct answers?

n = 11p = 0.2k = 5

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.8501. Find z to two decimals.

A = 0.8501

z =

6.Normal distributionHard

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. Find the value x below which 95% of them lie (the 95th percentile).

μ = 32σ = 6P = 95%

x = min

7.Confidence intervalsMedium

The blood pressure of 13 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 5.8 and standard deviation 6.3. Find the margin of error E of the 95% confidence interval for the mean difference.

n = 13d̄ = 5.8s_d = 6.395%t(12) = 2.179

E = mmHg

8.Confidence intervalsHard

Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 90% confidence interval for the proportion has a margin of error of at most 0.04?

p = 0.5E = 0.0490%

n =

9.Hypothesis tests & confidence intervalsMedium

Two teaching methods are compared. Group 1: 13 students, mean score 111.5, standard deviation 2.5. Group 2: 8 students, mean score 104.4, standard deviation 8. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).

n₁ = 13x̄₁ = 111.5s₁ = 2.5n₂ = 8x̄₂ = 104.4s₂ = 8

t =

10.Hypothesis tests & confidence intervalsHard

How large must a sample be so that the 95% confidence interval for a mean has a margin of error of at most 1.5? The standard deviation is 8. Find the smallest sample size n.

σ = 8E = 1.595%

n =

11.Linear regression & correlationMedium

The table gives 6 paired observations of x = car age in years and y = price in k€. Find the coefficient of determination r² of the least-squares line.

n = 6

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. Predict ŷ for x = 3.

n = 6x = 3

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 10.31

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 = 163/5 = 32.6

Squared deviations

Σ(x − x̄)² = (−13.6)² + 3.4² + 0.4² + (−4.6)² + 14.4² = 425.2

Standard deviation

s = √(425.2/4) = 10.31

Answer

s = 10.31

2.Descriptive statistics

Answer: P = 0.3085

Standardise

z = −0.5

Standard normal table

Φ(−0.5) = 0.3085

P = 0.3085

Answer

P = 0.3085

3.Probability

Answer: P = 0.04545

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.

First and second draw

P = 3/12 · 2/11 = 6/132 = 0.04545

Answer

P = 0.04545

4.Probability

Answer: P = 0.03876

Binomial formula

P(X = 5) = 462·0.2⁵·0.8⁶ = 462·0.00032·0.2621 = 0.03876

Answer

P = 0.03876

5.Normal distribution

Answer: z = 1.44

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 + 0.8501)/2 = 0.9251

Read the z-table backwards

Find 0.9251 inside the table and read off z.

z = 1.44

Answer

z = 1.44

6.Normal distribution

Answer: x = 41.87 min

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 = 32 + 1.645·6 = 41.87 min

Answer

x = 41.87 min

7.Confidence intervals

Answer: E = 3.807 mmHg

Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.

Critical value

df = 12, t = 2.179

Margin of error

E = 2.179·6.3/√13 = 3.807

Answer

E = 3.807 mmHg

8.Confidence intervals

Answer: n = 423

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 = 1.645²·0.5·0.5/0.04² = 422.8

Round up

n = 423

Answer

n = 423

9.Hypothesis tests & confidence intervals

Answer: t = 3.012

Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.

Pooled variance

s_p² = (12·2.5² + 7·8²)/19 = 27.53

s_p = 5.247

Standard error

SE = 5.247·√(1/13 + 1/8) = 2.358

Test statistic

t = (111.5 − 104.4)/2.358 = 3.012

Answer

t = 3.012

With df = 19, the two-sided 5% critical value is 2.093; |t| is larger, so H₀ is rejected.

10.Hypothesis tests & confidence intervals

Answer: n = 110

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).

Solve for n

n = (1.96·8/1.5)² = 109.3

Round up

n = 110

Answer

n = 110

11.Linear regression & correlation

Answer: r² = 0.9212

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 = 21/6 = 3.5

ȳ = 97/6 = 16.17

Sums of squares and products

S_xx = Σ(x − x̄)² = 17.5

S_xy = Σ(x − x̄)(y − ȳ) = −44.5

S_yy = Σ(y − ȳ)² = 122.8

Coefficient of determination

r² = −44.5²/(17.5·122.8) = 0.9212

Answer

r² = 0.9212

About 92.12% of the variation in y is explained by the line.

12.Linear regression & correlation

Answer: ŷ = 53.61

Means

x̄ = 42/6 = 7

ȳ = 430/6 = 71.67

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 316

Regression line

b = 316/70 = 4.514

a = 71.67 − 4.514·7 = 40.07

ŷ = 40.07 + 4.514·x

Prediction

ŷ = 40.07 + 4.514·3 = 53.61

Answer

ŷ = 53.61

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