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: 11, 20, 54, 14, 19, 25, 36.

n = 7

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 50 and standard deviation 8. What proportion P of values is above 60?

μ = 50σ = 8x = 60

P =

3.ProbabilityMedium

A bag holds 8 red and 7 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?

8 / 7

P =

4.ProbabilityHard

A test has 7 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 = 7p = 0.2k = 5

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.8444. Find z to two decimals.

A = 0.8444

z =

6.Normal distributionHard

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Find the value x below which 95% of them lie (the 95th percentile).

μ = 250σ = 5P = 95%

x = g

7.Confidence intervalsMedium

The battery life of two phone models is compared. Model 1: 60 phones, mean 97 h, known standard deviation 12 h. Model 2: 50 phones, mean 106.5 h, known standard deviation 18 h. Find the upper limit of the 90% confidence interval for μ₁ − μ₂.

n₁ = 60x̄₁ = 97 hσ₁ = 12 hn₂ = 50x̄₂ = 106.5 hσ₂ = 18 h90%

upper limit = h

8.Confidence intervalsHard

The population standard deviation is 22. How large a sample n do you need so that a 90% confidence interval for the mean has a margin of error of at most 2?

σ = 22E = 290%

n =

9.Hypothesis tests & confidence intervalsMedium

The mean of a population is claimed to be 100; the population standard deviation is known to be 8. A random sample of 64 observations has a mean of 101.32. Test H₀: μ = 100 against H₁: μ > 100. Find the p-value p.

μ₀ = 100σ = 8n = 64x̄ = 101.32H₁: μ > μ₀

p =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.01 must detect a shift of the mean of 4 units with a power of 80%. The standard deviation is 23. Find the minimum sample size n.

α = 0.01power = 80%δ = 4σ = 23z_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€. Find Pearson's correlation coefficient r.

n = 5

r =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. A new observation (11, 35) is made. Find its residual e = y − ŷ.

n = 6(11, 35)

e =

3Solutions

1.Descriptive statistics

Answer: s = 14.93

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 = 179/7 = 25.57

Squared deviations

Σ(x − x̄)² = (−14.57)² + (−5.571)² + 28.43² + (−11.57)² + (−6.571)² + (−0.5714)² + 10.43² = 1338

Standard deviation

s = √(1338/6) = 14.93

Answer

s = 14.93

2.Descriptive statistics

Answer: P = 0.1056

Standardise

z = 1.25

Standard normal table

Φ(1.25) = 0.8944

P = 1 − 0.8944 = 0.1056

Answer

P = 0.1056

3.Probability

Answer: P = 0.2667

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 = 8/15 · 7/14 = 56/210 = 0.2667

Answer

P = 0.2667

4.Probability

Answer: P = 0.004301

Binomial formula

P(X = 5) = 21·0.2⁵·0.8² = 21·0.00032·0.64 = 0.004301

Answer

P = 0.004301

5.Normal distribution

Answer: z = 1.42

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 + 0.8444)/2 = 0.9222

Read the z-table backwards

Find 0.9222 inside the table and read off z.

z = 1.42

Answer

z = 1.42

6.Normal distribution

Answer: x = 258.2 g

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 = 250 + 1.645·5 = 258.22 g

Answer

x = 258.22 g

7.Confidence intervals

Answer: upper limit = −4.598 h

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.

Point estimate

x̄₁ − x̄₂ = 97 − 106.5 = −9.5

Standard error

SE = √(12²/60 + 18²/50) = 2.98

Margin of error

E = 1.645·2.98 = 4.902

Interval

−9.5 ± 4.902 → [−14.4, −4.598]

Answer

upper limit = −4.598 h

The interval does not contain 0, so the means differ significantly.

8.Confidence intervals

Answer: n = 328

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.

Solve for n

n = (1.645·22/2)² = 327.4

Round up

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

n = 328

Answer

n = 328

9.Hypothesis tests & confidence intervals

Answer: p = 0.09342

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.

Test statistic

SE = σ/√n = 8/√64 = 1

z = (101.32 − 100)/1 = 1.32

Tail area

Φ(1.32) = 0.9066

1 − Φ(1.32) = 0.0934

p-value

p = 0.09342

Answer

p = 0.09342

p > 0.05, so H₀ is not rejected at the 5% level.

10.Hypothesis tests & confidence intervals

Answer: n = 332

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)·23/4)² = 331.8

Round up

n = 332

Answer

n = 332

11.Linear regression & correlation

Answer: r = −0.8781

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 = 108/5 = 21.6

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = −16

S_yy = Σ(y − ȳ)² = 33.2

Correlation coefficient

r = −16/√(10·33.2) = −0.8781

Answer

r = −0.8781

12.Linear regression & correlation

Answer: e = −1.8

Means

x̄ = 42/6 = 7

ȳ = 144/6 = 24

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 224

Regression line

b = 224/70 = 3.2

a = 24 − 3.2·7 = 1.6

ŷ = 1.6 + 3.2·x

Predicted value

ŷ = 1.6 + 3.2·11 = 36.8

Residual

e = 35 − 36.8 = −1.8

Answer

e = −1.8

The point lies below the line.

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