Introductory Statistics: cram pack

Formula sheets, the key method steps for every topic and 12 practice problems, with full solutions at the back.

New problems Back to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

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: 54, 33, 33, 54, 38.

n = 5

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 170 and standard deviation 12. Find the z-score z of a score of 174.8.

μ = 170σ = 12x = 174.8

z =

3.ProbabilityMedium

For the distribution shown, find the variance Var(X) of X.

x: 0, 1, 2, 3, 4p(x): 0.2, 0.2, 0.05, 0.3, 0.25

Var(X) =

4.ProbabilityHard

11 components are tested, and each one fails with probability 0.2, independently. What is the probability P that at most one fails?

n = 11p = 0.2

P =

5.Normal distributionMedium

The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. What fraction lies between 164 and 181.85 cm? Give P as a decimal.

μ = 178σ = 7x₁ = 164x₂ = 181.85

P =

6.Normal distributionHard

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Find the value x below which 80% of them lie (the 80th percentile).

μ = 250σ = 5P = 80%

x = g

7.Confidence intervalsMedium

A sample of 12 items has mean 88.5 and standard deviation s = 14.5. The population is roughly normal. Find the margin of error E of a 95% confidence interval for the mean.

n = 12x̄ = 88.5s = 14.595%t(11) = 2.201

E =

8.Confidence intervalsHard

The population standard deviation is 10. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 4?

σ = 10E = 499%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 119.1 and a sample standard deviation of 5. Find the test statistic t for H₀: μ = 120.

μ₀ = 120x̄ = 119.1s = 5n = 25

t =

10.Hypothesis tests & confidence intervalsHard

A sample of 49 measurements has mean 49.8. The population standard deviation is 12. Find the upper limit U of the 95% confidence interval for the mean.

x̄ = 49.8σ = 12n = 4995%

U =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. 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 = car age in years and y = price in k€. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 22?

n = 6ŷ = 22

x =

3Solutions

1.Descriptive statistics

Answer: s = 10.78

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 = 212/5 = 42.4

Squared deviations

Σ(x − x̄)² = 11.6² + (−9.4)² + (−9.4)² + 11.6² + (−4.4)² = 465.2

Standard deviation

s = √(465.2/4) = 10.78

Answer

s = 10.78

2.Descriptive statistics

Answer: z = 0.4

Standardise

z = (174.8 − 170)/12 = 0.4

Answer

z = 0.4

3.Probability

Answer: Var(X) = 2.26

Discrete random variables: a probability distribution lists every value x a variable can take with its probability p(x); the probabilities are between 0 and 1 and add up to 1. The expected value (mean) is E(X) = Σx·p(x), the long-run average. The variance is Var(X) = Σ(x − μ)²·p(x) = E(X²) − μ², and the standard deviation σ = √Var(X). For a linear change Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X) and σ_Y = |a|·σ_X; adding a constant does not change the spread.

Mean

μ = 0·0.2 + 1·0.2 + 2·0.05 + 3·0.3 + 4·0.25 = 2.2

Mean of the squares

E(X²) = 0·0.2 + 1·0.2 + 4·0.05 + 9·0.3 + 16·0.25 = 7.1

Variance

Var(X) = 7.1 − 2.2² = 2.26

Answer

Var(X) = 2.26

4.Probability

Answer: P = 0.3221

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.

Zero and one failure

P(X = 0) = 0.8¹¹ = 0.0859

P(X = 1) = 11·0.2·0.8¹⁰ = 0.2362

Add them

P = 0.0859 + 0.2362 = 0.3221

Answer

P = 0.3221

5.Normal distribution

Answer: P = 0.6861

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 = (164 − 178)/7 = −2

z₂ = (181.85 − 178)/7 = 0.55

Area between

P = Φ(0.55) − Φ(−2) = 0.7088 − 0.02275 = 0.6861

Answer

P = 0.6861

6.Normal distribution

Answer: x = 254.2 g

z for the percentile

Φ(z) = 0.8 → z = 0.8416

Back to x

x = 250 + 0.8416·5 = 254.21 g

Answer

x = 254.21 g

7.Confidence intervals

Answer: E = 9.213

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 = 11, t = 2.201

Margin of error

E = 2.201·14.5/√12 = 9.213

Answer

E = 9.213

8.Confidence intervals

Answer: n = 42

Solve for n

n = (2.576·10/4)² = 41.47

Round up

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

n = 42

Answer

n = 42

9.Hypothesis tests & confidence intervals

Answer: t = −0.9

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).

Standard error

SE = 5/√25 = 5/5 = 1

Test statistic

t = (119.1 − 120)/1 = −0.9

Answer

t = −0.9

t_{0.025; df = 24} = 2.064

|t| < 2.064, so H₀ is not rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: U = 53.16

Margin of error

E = 1.96·12/√49 = 3.36

Interval limit

U = 49.8 + 3.36 = 53.16

Answer

U = 53.16

11.Linear regression & correlation

Answer: r² = 0.942

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 = 25/5 = 5

ȳ = 214/5 = 42.8

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 128

S_yy = Σ(y − ȳ)² = 434.8

Coefficient of determination

r² = 128²/(40·434.8) = 0.942

Answer

r² = 0.942

About 94.2% of the variation in y is explained by the line.

12.Linear regression & correlation

Answer: x = 1.629

Means

x̄ = 21/6 = 3.5

ȳ = 98/6 = 16.33

Sums of squares and products

S_xx = Σ(x − x̄)² = 17.5

S_xy = Σ(x − x̄)(y − ȳ) = −53

Regression line

b = −53/17.5 = −3.029

a = 16.33 − (−3.029)·3.5 = 26.93

ŷ = 26.93 − 3.029·x

Solve for x

x = (22 − 26.93)/(−3.029) = 1.629

Answer

x = 1.629

ExamProblems · examproblems.com/courses/statistics-1