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: 48, 51, 32, 58, 37.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 70 and standard deviation 20. What proportion P of values is below 50?

μ = 70σ = 20x = 50

P =

3.ProbabilityMedium

In how many ways N can a committee of 3 people be chosen from a group of 11? The order does not matter.

n = 11k = 3

N =

4.ProbabilityHard

9 components are tested, and each one fails with probability 0.25, independently. What is the probability P that at most one fails?

n = 9p = 0.25

P =

5.Normal distributionMedium

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. What fraction lies between 24.8 and 44.3 min? Give P as a decimal.

μ = 32σ = 6x₁ = 24.8x₂ = 44.3

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 30% is to the left of it. Find the cut-off x.

μ = 250 gσ = 5 g30%

x = g

7.Confidence intervalsMedium

A 99% confidence interval for a mean, based on 36 observations and a known σ, has a margin of error of 9.2. What is the margin of error E₂ of a 90% interval from the same data?

99%n₁ = 36E₁ = 9.290%

E₂ =

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 0.5?

σ = 26E = 0.590%

n =

9.Hypothesis tests & confidence intervalsMedium

Two teaching methods are compared. Group 1: 8 students, mean score 31.5, standard deviation 7.5. Group 2: 9 students, mean score 27.5, standard deviation 7.5. 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 = 8x̄₁ = 31.5s₁ = 7.5n₂ = 9x̄₂ = 27.5s₂ = 7.5

t =

10.Hypothesis tests & confidence intervalsHard

In a trial, 50 of 100 patients on treatment A recover, and 162 of 300 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.

x₁ = 50n₁ = 100x₂ = 162n₂ = 300

z =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = load in kN and y = extension in mm. Find the coefficient of determination r² of the least-squares line.

n = 5

r² =

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 = 91?

n = 6ŷ = 91

x =

3Solutions

1.Descriptive statistics

Answer: s = 10.57

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 = 226/5 = 45.2

Squared deviations

Σ(x − x̄)² = 2.8² + 5.8² + (−13.2)² + 12.8² + (−8.2)² = 446.8

Standard deviation

s = √(446.8/4) = 10.57

Answer

s = 10.57

2.Descriptive statistics

Answer: P = 0.1587

Standardise

z = −1

Standard normal table

Φ(−1) = 0.1587

P = 0.1587

Answer

P = 0.1587

3.Probability

Answer: N = 165

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.

Combinations

C(11, 3) = (11·10·9)/(3·2·1) = 165

Answer

N = 165

4.Probability

Answer: P = 0.3003

Zero and one failure

P(X = 0) = 0.75⁹ = 0.07508

P(X = 1) = 9·0.25·0.75⁸ = 0.2253

Add them

P = 0.07508 + 0.2253 = 0.3003

Answer

P = 0.3003

5.Normal distribution

Answer: P = 0.8647

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 = (24.8 − 32)/6 = −1.2

z₂ = (44.3 − 32)/6 = 2.05

Area between

P = Φ(2.05) − Φ(−1.2) = 0.9798 − 0.1151 = 0.8647

Answer

P = 0.8647

6.Normal distribution

Answer: x = 247.4 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) = 0.3

z from the table

z ≈ −0.52

Back to the original units

x = 250 − 0.52·5 = 247.4 g

Answer

x = 247.4 g

7.Confidence intervals

Answer: E₂ = 5.875

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 = (1.645/2.576)·√(36/36) = 0.6386

New margin

E₂ = 9.2·0.6386 = 5.875

Answer

E₂ = 5.875

8.Confidence intervals

Answer: n = 7318

Solve for n

n = (1.645·26/0.5)² = 7317

Round up

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

n = 7318

Answer

n = 7318

9.Hypothesis tests & confidence intervals

Answer: t = 1.098

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 = (7·7.5² + 8·7.5²)/15 = 56.25

s_p = 7.5

Standard error

SE = 7.5·√(1/8 + 1/9) = 3.644

Test statistic

t = (31.5 − 27.5)/3.644 = 1.098

Answer

t = 1.098

With df = 15, the two-sided 5% critical value is 2.131; |t| is smaller, so H₀ is not rejected.

10.Hypothesis tests & confidence intervals

Answer: z = −0.6941

Sample proportions

p̂₁ = 50/100 = 0.5

p̂₂ = 162/300 = 0.54

p̂ = 212/400 = 0.53

Standard error under H₀

SE = √(0.53·0.47·(1/100 + 1/300)) = 0.05763

Test statistic

z = (0.5 − 0.54)/0.05763 = −0.6941

Answer

z = −0.6941

|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).

11.Linear regression & correlation

Answer: r² = 0.9981

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 = 30/5 = 6

ȳ = 123/5 = 24.6

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 160

S_yy = Σ(y − ȳ)² = 641.2

Coefficient of determination

r² = 160²/(40·641.2) = 0.9981

Answer

r² = 0.9981

About 99.81% of the variation in y is explained by the line.

12.Linear regression & correlation

Answer: x = 25.72

Means

x̄ = 153/6 = 25.5

ȳ = 541/6 = 90.17

Sums of squares and products

S_xx = Σ(x − x̄)² = 157.5

S_xy = Σ(x − x̄)(y − ȳ) = 595.5

Regression line

b = 595.5/157.5 = 3.781

a = 90.17 − 3.781·25.5 = −6.248

ŷ = −6.248 + 3.781·x

Solve for x

x = (91 − (−6.248))/3.781 = 25.72

Answer

x = 25.72

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