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: 12, 30, 31, 30, 22, 21, 45.

n = 7

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 100 and standard deviation 20. Find the z-score z of a score of 76.

μ = 100σ = 20x = 76

z =

3.ProbabilityMedium

A fair die is rolled 4 times. What is the probability P of getting at least one six?

n = 4

P =

4.ProbabilityHard

A test has 8 multiple-choice questions. A student who guesses gets each one right with probability 0.25, independently. What is the probability P of exactly 5 correct answers?

n = 8p = 0.25k = 5

P =

5.Normal distributionMedium

Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.2005. Find z to two decimals.

A = 0.2005

z =

6.Normal distributionHard

The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. Find the value x below which 75% of them lie (the 75th percentile).

μ = 178σ = 7P = 75%

x = cm

7.Confidence intervalsMedium

The blood pressure of 13 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 3.3 and standard deviation 8. Find the lower limit of the 95% confidence interval for the mean difference.

n = 13d̄ = 3.3s_d = 895%t(12) = 2.179

lower limit = mmHg

8.Confidence intervalsHard

The population standard deviation is 28. 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 1?

σ = 28E = 195%

n =

9.Hypothesis tests & confidence intervalsMedium

Two production lines make the same part. Line 1: a sample of 80 parts has mean length 110 mm, and the line's known standard deviation is 3 mm. Line 2: a sample of 40 parts has mean 108 mm, known standard deviation 6 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).

n₁ = 80x̄₁ = 110 mmσ₁ = 3 mmn₂ = 40x̄₂ = 108 mmσ₂ = 6 mm

z =

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? The standard deviation is 4. Find the smallest sample size n.

σ = 4E = 195%

n =

11.Linear regression & correlationMedium

The table gives 6 paired observations of x = advertising spend in k€ and y = sales in k€. Their sums are given. Find Pearson's correlation coefficient r from these sums.

n = 6Σx = 48Σy = 445Σx² = 454Σy² = 35839Σxy = 4003

r =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 154?

n = 6ŷ = 154

x =

3Solutions

1.Descriptive statistics

Answer: s = 10.36

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 = 191/7 = 27.29

Squared deviations

Σ(x − x̄)² = (−15.29)² + 2.714² + 3.714² + 2.714² + (−5.286)² + (−6.286)² + 17.71² = 643.4

Standard deviation

s = √(643.4/6) = 10.36

Answer

s = 10.36

2.Descriptive statistics

Answer: z = −1.2

Standardise

z = (76 − 100)/20 = −1.2

Answer

z = −1.2

3.Probability

Answer: P = 0.5177

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.

Complement

P(no six) = (5/6)⁴ = 0.4823

P = 1 − 0.4823 = 0.5177

Answer

P = 0.5177

4.Probability

Answer: P = 0.02307

Binomial formula

P(X = 5) = 56·0.25⁵·0.75³ = 56·0.0009766·0.4219 = 0.02307

Answer

P = 0.02307

5.Normal distribution

Answer: z = 0.84

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 − 0.2005 = 0.7995

Read the z-table backwards

Find 0.7995 inside the table and read off z.

z = 0.84

Answer

z = 0.84

6.Normal distribution

Answer: x = 182.7 cm

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) = 0.75 → z = 0.6745

Back to x

x = 178 + 0.6745·7 = 182.72 cm

Answer

x = 182.72 cm

7.Confidence intervals

Answer: lower limit = −1.535 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 = 2.179·8/√13 = 4.835

Interval

3.3 ± 4.835 → [−1.535, 8.135]

Answer

lower limit = −1.535 mmHg

8.Confidence intervals

Answer: n = 3012

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 = (1.96·28/1)² = 3012

Round up

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

n = 3012

Answer

n = 3012

9.Hypothesis tests & confidence intervals

Answer: z = 1.988

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.

Standard error of the difference

SE = √(3²/80 + 6²/40) = √1.012 = 1.006

Test statistic

z = (110 − 108)/1.006 = 1.988

Answer

z = 1.988

|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: n = 62

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 = (1.96·4/1)² = 61.47

Round up

n = 62

Answer

n = 62

11.Linear regression & correlation

Answer: r = 0.9945

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.

Centred sums

S_xy = 4003 − 48·445/6 = 443

S_xx = 454 − 48²/6 = 70

S_yy = 35839 − 445²/6 = 2835

Correlation coefficient

r = 443/√(70·2835) = 0.9945

Answer

r = 0.9945

12.Linear regression & correlation

Answer: x = 27.43

Means

x̄ = 129/6 = 21.5

ȳ = 712/6 = 118.7

Sums of squares and products

S_xx = Σ(x − x̄)² = 157.5

S_xy = Σ(x − x̄)(y − ȳ) = 939

Regression line

b = 939/157.5 = 5.962

a = 118.7 − 5.962·21.5 = −9.514

ŷ = −9.514 + 5.962·x

Solve for x

x = (154 − (−9.514))/5.962 = 27.43

Answer

x = 27.43

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