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: 57, 54, 16, 14, 54, 28.

n = 6

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 500 and standard deviation 40. Find the z-score z of a score of 580.

μ = 500σ = 40x = 580

z =

3.ProbabilityMedium

A bag holds 8 red and 8 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?

8 / 8

P =

4.ProbabilityHard

8 components are tested, and each one fails with probability 0.7, independently. What is the probability P that at most one fails?

n = 8p = 0.7

P =

5.Normal distributionMedium

Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.9591. Find z to two decimals.

A = 0.9591

z =

6.Normal distributionHard

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Find the value x below which 20% of them lie (the 20th percentile).

μ = 65σ = 10P = 20%

x = points

7.Confidence intervalsMedium

A sample of 20 items has mean 92 and standard deviation s = 11. The population is roughly normal. Find the upper limit of the 95% confidence interval for the mean.

n = 20x̄ = 92s = 1195%t(19) = 2.093

upper limit =

8.Confidence intervalsHard

The population standard deviation is 39. 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 1.5?

σ = 39E = 1.595%

n =

9.Hypothesis tests & confidence intervalsMedium

The mean of a population is claimed to be 80; the population standard deviation is known to be 15. A random sample of 36 observations has a mean of 77.75. Test H₀: μ = 80 against H₁: μ < 80. Find the p-value p.

μ₀ = 80σ = 15n = 36x̄ = 77.75H₁: μ < μ₀

p =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test of H₀: μ = 200 against H₁: μ < 200 is done at α = 0.10 with a sample of 64; the standard deviation is known to be 20. In reality the mean is 193.5. Find the power 1 − β of the test in percent.

μ₀ = 200μ = 193.5σ = 20n = 64α = 0.10

1 − β = %

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 6 paired observations of x = hours studied and y = exam score. Use the least-squares line ŷ = a + b·x. A new observation (2.5, 62) is made. Find its residual e = y − ŷ.

n = 6(2.5, 62)

e =

3Solutions

1.Descriptive statistics

Answer: s = 20.14

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 = 223/6 = 37.17

Squared deviations

Σ(x − x̄)² = 19.83² + 16.83² + (−21.17)² + (−23.17)² + 16.83² + (−9.167)² = 2029

Standard deviation

s = √(2029/5) = 20.14

Answer

s = 20.14

2.Descriptive statistics

Answer: z = 2

Standardise

z = (580 − 500)/40 = 2

Answer

z = 2

3.Probability

Answer: P = 0.2333

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 = 8/16 · 7/15 = 56/240 = 0.2333

Answer

P = 0.2333

4.Probability

Answer: P = 0.00129

Zero and one failure

P(X = 0) = 0.3⁸ = 6.561 × 10⁻⁵

P(X = 1) = 8·0.7·0.3⁷ = 0.001225

Add them

P = 6.561 × 10⁻⁵ + 0.001225 = 0.00129

Answer

P = 0.00129

5.Normal distribution

Answer: z = −1.74

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 − 0.9591 = 0.0409

Read the z-table backwards

Find 0.0409 inside the table and read off z.

z = −1.74

Answer

z = −1.74

6.Normal distribution

Answer: x = 56.58 points

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) = 0.2 → z = −0.8416

Back to x

x = 65 + −0.8416·10 = 56.58 points

Answer

x = 56.58 points

7.Confidence intervals

Answer: upper limit = 97.15

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 = 19, t = 2.093

Margin of error

E = 2.093·11/√20 = 5.148

Interval

92 ± 5.148 → [86.85, 97.15]

Answer

upper limit = 97.15

8.Confidence intervals

Answer: n = 2597

Solve for n

n = (1.96·39/1.5)² = 2597

Round up

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

n = 2597

Answer

n = 2597

9.Hypothesis tests & confidence intervals

Answer: p = 0.1841

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 = 15/√36 = 2.5

z = (77.75 − 80)/2.5 = −0.9

Tail area

Φ(0.9) = 0.8159

1 − Φ(0.9) = 0.1841

p-value

p = 0.1841

Answer

p = 0.1841

p > 0.05, so H₀ is not rejected at the 5% level.

10.Hypothesis tests & confidence intervals

Answer: 1 − β = 90.62 %

Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).

Critical sample mean

SE = 20/√64 = 2.5

x̄_c = 200 − 1.282·2.5 = 196.8

z under the true mean

z = (196.8 − 193.5)/2.5 = 1.318

Power

power = P(Z < 1.318) = 90.62%

1 − β = 90.62%

Answer

1 − β = 90.62%

11.Linear regression & correlation

Answer: r = 0.9537

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 = 20/5 = 4

ȳ = 312/5 = 62.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = 57

S_yy = Σ(y − ȳ)² = 357.2

Correlation coefficient

r = 57/√(10·357.2) = 0.9537

Answer

r = 0.9537

12.Linear regression & correlation

Answer: e = 1.567

Means

x̄ = 27/6 = 4.5

ȳ = 413/6 = 68.83

Sums of squares and products

S_xx = Σ(x − x̄)² = 17.5

S_xy = Σ(x − x̄)(y − ȳ) = 73.5

Regression line

b = 73.5/17.5 = 4.2

a = 68.83 − 4.2·4.5 = 49.93

ŷ = 49.93 + 4.2·x

Predicted value

ŷ = 49.93 + 4.2·2.5 = 60.43

Residual

e = 62 − 60.43 = 1.567

Answer

e = 1.567

The point lies above the line.

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