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: 42, 30, 53, 39, 11.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 60 and standard deviation 12. What proportion P of values is above 50.4?

μ = 60σ = 12x = 50.4

P =

3.ProbabilityMedium

In how many ways N can a committee of 4 people be chosen from a group of 12? The order does not matter.

n = 12k = 4

N =

4.ProbabilityHard

7 components are tested, and each one fails with probability 0.4, independently. What is the probability P that at most one fails?

n = 7p = 0.4

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.6875. Find z to two decimals.

A = 0.6875

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 10% of them lie (the 10th percentile).

μ = 250σ = 5P = 10%

x = g

7.Confidence intervalsMedium

The battery life of two phone models is compared. Model 1: 50 phones, mean 52.5 h, known standard deviation 18 h. Model 2: 50 phones, mean 45 h, known standard deviation 19 h. Find the upper limit of the 90% confidence interval for μ₁ − μ₂.

n₁ = 50x̄₁ = 52.5 hσ₁ = 18 hn₂ = 50x̄₂ = 45 hσ₂ = 19 h90%

upper limit = h

8.Confidence intervalsHard

The population standard deviation is 33. 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 0.5?

σ = 33E = 0.599%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 64 students has a mean score of 51.81 and a sample standard deviation of 5. Find the test statistic t for H₀: μ = 50.

μ₀ = 50x̄ = 51.81s = 5n = 64

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test of H₀: μ = 80 against H₁: μ < 80 is done at α = 0.05 with a sample of 100; the standard deviation is known to be 10. In reality the mean is 76.6. Find the power 1 − β of the test in percent.

μ₀ = 80μ = 76.6σ = 10n = 100α = 0.05

1 − β = %

11.Linear regression & correlationMedium

The table gives 5 paired observations of x = hours studied and y = exam score. Their sums are given. Find Pearson's correlation coefficient r from these sums.

n = 5Σx = 25Σy = 321Σx² = 165Σy² = 22197Σxy = 1853

r =

12.Linear regression & correlationHard

The table gives 6 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. A new observation (6, 71) is made. Find its residual e = y − ŷ.

n = 6(6, 71)

e =

3Solutions

1.Descriptive statistics

Answer: s = 15.73

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 = 175/5 = 35

Squared deviations

Σ(x − x̄)² = 7² + (−5)² + 18² + 4² + (−24)² = 990

Standard deviation

s = √(990/4) = 15.73

Answer

s = 15.73

2.Descriptive statistics

Answer: P = 0.7881

Standardise

z = −0.8

Standard normal table

Φ(−0.8) = 0.2119

P = 1 − 0.2119 = 0.7881

Answer

P = 0.7881

3.Probability

Answer: N = 495

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(12, 4) = (12·11·10·9)/(4·3·2·1) = 495

Answer

N = 495

4.Probability

Answer: P = 0.1586

Zero and one failure

P(X = 0) = 0.6⁷ = 0.02799

P(X = 1) = 7·0.4·0.6⁶ = 0.1306

Add them

P = 0.02799 + 0.1306 = 0.1586

Answer

P = 0.1586

5.Normal distribution

Answer: z = 1.01

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 + 0.6875)/2 = 0.8438

Read the z-table backwards

Find 0.8438 inside the table and read off z.

z = 1.01

Answer

z = 1.01

6.Normal distribution

Answer: x = 243.6 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) = 0.1 → z = −1.282

Back to x

x = 250 + −1.282·5 = 243.59 g

Answer

x = 243.59 g

7.Confidence intervals

Answer: upper limit = 13.59 h

Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.

Point estimate

x̄₁ − x̄₂ = 52.5 − 45 = 7.5

Standard error

SE = √(18²/50 + 19²/50) = 3.701

Margin of error

E = 1.645·3.701 = 6.089

Interval

7.5 ± 6.089 → [1.411, 13.59]

Answer

upper limit = 13.59 h

The interval does not contain 0, so the means differ significantly.

8.Confidence intervals

Answer: n = 28906

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.

Solve for n

n = (2.576·33/0.5)² = 28905

Round up

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

n = 28906

Answer

n = 28906

9.Hypothesis tests & confidence intervals

Answer: t = 2.896

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/√64 = 5/8 = 0.625

Test statistic

t = (51.81 − 50)/0.625 = 2.896

Answer

t = 2.896

t_{0.025; df = 63} = 1.998

|t| > 1.998, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: 1 − β = 96.04 %

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 = 10/√100 = 1

x̄_c = 80 − 1.645·1 = 78.36

z under the true mean

z = (78.36 − 76.6)/1 = 1.755

Power

power = P(Z < 1.755) = 96.04%

1 − β = 96.04%

Answer

1 − β = 96.04%

11.Linear regression & correlation

Answer: r = 0.9838

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.

Centred sums

S_xy = 1853 − 25·321/5 = 248

S_xx = 165 − 25²/5 = 40

S_yy = 22197 − 321²/5 = 1589

Correlation coefficient

r = 248/√(40·1589) = 0.9838

Answer

r = 0.9838

12.Linear regression & correlation

Answer: e = 4.69

Means

x̄ = 48/6 = 8

ȳ = 457/6 = 76.17

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 345

Regression line

b = 345/70 = 4.929

a = 76.17 − 4.929·8 = 36.74

ŷ = 36.74 + 4.929·x

Predicted value

ŷ = 36.74 + 4.929·6 = 66.31

Residual

e = 71 − 66.31 = 4.69

Answer

e = 4.69

The point lies above the line.

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