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: 26, 40, 11, 29, 43, 18, 50.

n = 7

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 70 and standard deviation 20. What proportion P of values is below 110?

μ = 70σ = 20x = 110

P =

3.ProbabilityMedium

In how many ways N can a committee of 3 people be chosen from a group of 7? The order does not matter.

n = 7k = 3

N =

4.ProbabilityHard

A test has 7 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 = 7p = 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 0.0179. Find z to two decimals.

A = 0.0179

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 80% of them lie (the 80th percentile).

μ = 178σ = 7P = 80%

x = cm

7.Confidence intervalsMedium

A sample of 11 items has mean 103.5 and standard deviation s = 10. The population is roughly normal. Find the upper limit of the 90% confidence interval for the mean.

n = 11x̄ = 103.5s = 1090%t(10) = 1.812

upper limit =

8.Confidence intervalsHard

The population standard deviation is 36. 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 0.5?

σ = 36E = 0.595%

n =

9.Hypothesis tests & confidence intervalsMedium

Two teaching methods are compared. Group 1: 7 students, mean score 64.5, standard deviation 3. Group 2: 24 students, mean score 61.6, standard deviation 4. 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 = 7x̄₁ = 64.5s₁ = 3n₂ = 24x̄₂ = 61.6s₂ = 4

t =

10.Hypothesis tests & confidence intervalsHard

In a trial, 50 of 100 patients on treatment A recover, and 186 of 300 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.

x₁ = 50n₁ = 100x₂ = 186n₂ = 300

z =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Find Pearson's correlation coefficient r.

n = 5

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. For which value x does the line predict ŷ = 113?

n = 5ŷ = 113

x =

3Solutions

1.Descriptive statistics

Answer: s = 14.05

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 = 217/7 = 31

Squared deviations

Σ(x − x̄)² = (−5)² + 9² + (−20)² + (−2)² + 12² + (−13)² + 19² = 1184

Standard deviation

s = √(1184/6) = 14.05

Answer

s = 14.05

2.Descriptive statistics

Answer: P = 0.9772

Standardise

z = 2

Standard normal table

Φ(2) = 0.9772

P = 0.9772

Answer

P = 0.9772

3.Probability

Answer: N = 35

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(7, 3) = (7·6·5)/(3·2·1) = 35

Answer

N = 35

4.Probability

Answer: P = 0.01154

Binomial formula

P(X = 5) = 21·0.25⁵·0.75² = 21·0.0009766·0.5625 = 0.01154

Answer

P = 0.01154

5.Normal distribution

Answer: z = 2.1

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 − 0.0179 = 0.9821

Read the z-table backwards

Find 0.9821 inside the table and read off z.

z = 2.10

Answer

z = 2.10

6.Normal distribution

Answer: x = 183.9 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) = 0.8 → z = 0.8416

Back to x

x = 178 + 0.8416·7 = 183.89 cm

Answer

x = 183.89 cm

7.Confidence intervals

Answer: upper limit = 109

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.

Critical value

df = n − 1 = 10, t = 1.812

Margin of error

E = 1.812·10/√11 = 5.463

Interval

103.5 ± 5.463 → [98.04, 109]

Answer

upper limit = 109

8.Confidence intervals

Answer: n = 19915

Solve for n

n = (1.96·36/0.5)² = 19915

Round up

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

n = 19915

Answer

n = 19915

9.Hypothesis tests & confidence intervals

Answer: t = 1.77

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 = (6·3² + 23·4²)/29 = 14.55

s_p = 3.815

Standard error

SE = 3.815·√(1/7 + 1/24) = 1.639

Test statistic

t = (64.5 − 61.6)/1.639 = 1.77

Answer

t = 1.77

With df = 29, the two-sided 5% critical value is 2.045; |t| is smaller, so H₀ is not rejected.

10.Hypothesis tests & confidence intervals

Answer: z = −2.113

Sample proportions

p̂₁ = 50/100 = 0.5

p̂₂ = 186/300 = 0.62

p̂ = 236/400 = 0.59

Standard error under H₀

SE = √(0.59·0.41·(1/100 + 1/300)) = 0.05679

Test statistic

z = (0.5 − 0.62)/0.05679 = −2.113

Answer

z = −2.113

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

11.Linear regression & correlation

Answer: r = 0.9948

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 = 105/5 = 21

ȳ = 368/5 = 73.6

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 204

S_yy = Σ(y − ȳ)² = 1051

Correlation coefficient

r = 204/√(40·1051) = 0.9948

Answer

r = 0.9948

12.Linear regression & correlation

Answer: x = 25.4

Means

x̄ = 105/5 = 21

ȳ = 466/5 = 93.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 90

S_xy = Σ(x − x̄)(y − ȳ) = 405

Regression line

b = 405/90 = 4.5

a = 93.2 − 4.5·21 = −1.3

ŷ = −1.3 + 4.5·x

Solve for x

x = (113 − (−1.3))/4.5 = 25.4

Answer

x = 25.4

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