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: 58, 39, 59, 35, 57, 30, 17.

n = 7

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 100 and standard deviation 12. What proportion P of values is below 115?

μ = 100σ = 12x = 115

P =

3.ProbabilityMedium

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

7 / 7

P =

4.ProbabilityHard

6 components are tested, and each one fails with probability 0.4, independently. What is the probability P that at most one fails?

n = 6p = 0.4

P =

5.Normal distributionMedium

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. What fraction lies between 246 and 260.25 g? Give P as a decimal.

μ = 250σ = 5x₁ = 246x₂ = 260.25

P =

6.Normal distributionHard

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Drag the cut-off so the shaded bottom 20% is to the left of it. Find the cut-off x.

μ = 250 gσ = 5 g20%

x = g

7.Confidence intervalsMedium

The blood pressure of 10 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 9 and standard deviation 3.1. Find the lower limit of the 99% confidence interval for the mean difference.

n = 10d̄ = 9s_d = 3.199%t(9) = 3.25

lower limit = mmHg

8.Confidence intervalsHard

The population standard deviation is 26. 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 2.5?

σ = 26E = 2.590%

n =

9.Hypothesis tests & confidence intervalsMedium

The mean of a population is claimed to be 50; the population standard deviation is known to be 12. A random sample of 144 observations has a mean of 48.2. Test H₀: μ = 50 against H₁: μ < 50. Find the p-value p.

μ₀ = 50σ = 12n = 144x̄ = 48.2H₁: μ < μ₀

p =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.05 must detect a shift of the mean of 3 units with a power of 95%. The standard deviation is 16. Find the minimum sample size n.

α = 0.05power = 95%δ = 3σ = 16z_α = 1.645z_β = 1.645

n =

11.Linear regression & correlationMedium

The table gives 6 paired observations of x = temperature in °C and y = ice creams sold. Find Pearson's correlation coefficient r.

n = 6

r =

12.Linear regression & correlationHard

The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 15.5.

n = 5x = 15.5

ŷ =

3Solutions

1.Descriptive statistics

Answer: s = 16.31

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 = 295/7 = 42.14

Squared deviations

Σ(x − x̄)² = 15.86² + (−3.143)² + 16.86² + (−7.143)² + 14.86² + (−12.14)² + (−25.14)² = 1597

Standard deviation

s = √(1597/6) = 16.31

Answer

s = 16.31

2.Descriptive statistics

Answer: P = 0.8944

Standardise

z = 1.25

Standard normal table

Φ(1.25) = 0.8944

P = 0.8944

Answer

P = 0.8944

3.Probability

Answer: P = 0.2308

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 = 7/14 · 6/13 = 42/182 = 0.2308

Answer

P = 0.2308

4.Probability

Answer: P = 0.2333

Zero and one failure

P(X = 0) = 0.6⁶ = 0.04666

P(X = 1) = 6·0.4·0.6⁵ = 0.1866

Add them

P = 0.04666 + 0.1866 = 0.2333

Answer

P = 0.2333

5.Normal distribution

Answer: P = 0.768

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-scores

z₁ = (246 − 250)/5 = −0.8

z₂ = (260.25 − 250)/5 = 2.05

Area between

P = Φ(2.05) − Φ(−0.8) = 0.9798 − 0.2119 = 0.768

Answer

P = 0.768

6.Normal distribution

Answer: x = 245.8 g

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

Φ(z) = 0.2

z from the table

z ≈ −0.84

Back to the original units

x = 250 − 0.84·5 = 245.8 g

Answer

x = 245.8 g

7.Confidence intervals

Answer: lower limit = 5.814 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 = 9, t = 3.25

Margin of error

E = 3.25·3.1/√10 = 3.186

Interval

9 ± 3.186 → [5.814, 12.19]

Answer

lower limit = 5.814 mmHg

8.Confidence intervals

Answer: n = 293

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 = (1.645·26/2.5)² = 292.7

Round up

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

n = 293

Answer

n = 293

9.Hypothesis tests & confidence intervals

Answer: p = 0.03593

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 = 12/√144 = 1

z = (48.2 − 50)/1 = −1.8

Tail area

Φ(1.8) = 0.9641

1 − Φ(1.8) = 0.0359

p-value

p = 0.03593

Answer

p = 0.03593

p ≤ 0.05, so H₀ is rejected at the 5% level.

10.Hypothesis tests & confidence intervals

Answer: n = 308

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 = ((1.645 + 1.645)·16/3)² = 307.9

Round up

n = 308

Answer

n = 308

11.Linear regression & correlation

Answer: r = 0.9875

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 = 153/6 = 25.5

ȳ = 662/6 = 110.3

Sums of squares and products

S_xx = Σ(x − x̄)² = 157.5

S_xy = Σ(x − x̄)(y − ȳ) = 813

S_yy = Σ(y − ȳ)² = 4303

Correlation coefficient

r = 813/√(157.5·4303) = 0.9875

Answer

r = 0.9875

12.Linear regression & correlation

Answer: ŷ = 34.5

Means

x̄ = 100/5 = 20

ȳ = 237/5 = 47.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 90

S_xy = Σ(x − x̄)(y − ȳ) = 258

Regression line

b = 258/90 = 2.867

a = 47.4 − 2.867·20 = −9.933

ŷ = −9.933 + 2.867·x

Prediction

ŷ = −9.933 + 2.867·15.5 = 34.5

Answer

ŷ = 34.5

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