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: 20, 52, 45, 37, 38.

n = 5

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 70 and standard deviation 15. Find the z-score z of a score of 100.

μ = 70σ = 15x = 100

z =

3.ProbabilityMedium

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

x: 0, 2, 4, 6p(x): 0.35, 0.3, 0.25, 0.1

Var(X) =

4.ProbabilityHard

X has the distribution shown, and Y = −2X − 3. Find the standard deviation σ_Y of Y.

x: 1, 2, 3, 4, 5p(x): 0.2, 0.35, 0.15, 0.25, 0.05Y = −2X − 3

σ_Y =

5.Normal distributionMedium

Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.0793. Find z to two decimals.

A = 0.0793

z =

6.Normal distributionHard

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Find the value x below which 90% of them lie (the 90th percentile).

μ = 250σ = 5P = 90%

x = g

7.Confidence intervalsMedium

A sample of 25 items has mean 86 and standard deviation s = 15.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.

n = 25x̄ = 86s = 15.599%t(24) = 2.797

E =

8.Confidence intervalsHard

The population standard deviation is 15. 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 2.5?

σ = 15E = 2.599%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 114.6 and a sample standard deviation of 15. Find the test statistic t for H₀: μ = 120.

μ₀ = 120x̄ = 114.6s = 15n = 25

t =

10.Hypothesis tests & confidence intervalsHard

In a trial, 135 of 300 patients on treatment A recover, and 29 of 80 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.

x₁ = 135n₁ = 300x₂ = 29n₂ = 80

z =

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Find Pearson's correlation coefficient r.

n = 5

r =

12.Linear regression & correlationHard

The table gives 5 paired observations of x = hours studied and y = exam score. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 73?

n = 5ŷ = 73

x =

3Solutions

1.Descriptive statistics

Answer: s = 11.93

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 = 192/5 = 38.4

Squared deviations

Σ(x − x̄)² = (−18.4)² + 13.6² + 6.6² + (−1.4)² + (−0.4)² = 569.2

Standard deviation

s = √(569.2/4) = 11.93

Answer

s = 11.93

2.Descriptive statistics

Answer: z = 2

Standardise

z = (100 − 70)/15 = 2

Answer

z = 2

3.Probability

Answer: Var(X) = 3.96

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 = 0·0.35 + 2·0.3 + 4·0.25 + 6·0.1 = 2.2

Mean of the squares

E(X²) = 0·0.35 + 4·0.3 + 16·0.25 + 36·0.1 = 8.8

Variance

Var(X) = 8.8 − 2.2² = 3.96

Answer

Var(X) = 3.96

4.Probability

Answer: σ_Y = 2.4

Spread of X

μ = 2.6, Var(X) = 1.44, σ_X = 1.2

Linear change

σ_Y = |−2|·1.2 = 2.4

Answer

σ_Y = 2.4

The mean does move: E(Y) = −2·2.6 − 3 = −8.2.

5.Normal distribution

Answer: z = 1.41

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 − 0.0793 = 0.9207

Read the z-table backwards

Find 0.9207 inside the table and read off z.

z = 1.41

Answer

z = 1.41

6.Normal distribution

Answer: x = 256.4 g

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) = 0.9 → z = 1.282

Back to x

x = 250 + 1.282·5 = 256.41 g

Answer

x = 256.41 g

7.Confidence intervals

Answer: E = 8.671

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 = 24, t = 2.797

Margin of error

E = 2.797·15.5/√25 = 8.671

Answer

E = 8.671

8.Confidence intervals

Answer: n = 239

Solve for n

n = (2.576·15/2.5)² = 238.9

Round up

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

n = 239

Answer

n = 239

9.Hypothesis tests & confidence intervals

Answer: t = −1.8

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 = 15/√25 = 15/5 = 3

Test statistic

t = (114.6 − 120)/3 = −1.8

Answer

t = −1.8

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: z = 1.404

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.

Sample proportions

p̂₁ = 135/300 = 0.45

p̂₂ = 29/80 = 0.3625

p̂ = 164/380 = 0.4316

Standard error under H₀

SE = √(0.4316·0.5684·(1/300 + 1/80)) = 0.06232

Test statistic

z = (0.45 − 0.3625)/0.06232 = 1.404

Answer

z = 1.404

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

11.Linear regression & correlation

Answer: r = 0.9735

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 = 15/5 = 3

ȳ = 284/5 = 56.8

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = 63

S_yy = Σ(y − ȳ)² = 418.8

Correlation coefficient

r = 63/√(10·418.8) = 0.9735

Answer

r = 0.9735

12.Linear regression & correlation

Answer: x = 6.935

Means

x̄ = 30/5 = 6

ȳ = 347/5 = 69.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 154

Regression line

b = 154/40 = 3.85

a = 69.4 − 3.85·6 = 46.3

ŷ = 46.3 + 3.85·x

Solve for x

x = (73 − 46.3)/3.85 = 6.935

Answer

x = 6.935

ExamProblems · examproblems.com/courses/statistics-1