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: 19, 12, 33, 31, 21.

n = 5

s =

2.Descriptive statisticsHard

Exam scores are normally distributed with mean 170 and standard deviation 8. Find the z-score z of a score of 180.

μ = 170σ = 8x = 180

z =

3.ProbabilityMedium

A fair die is rolled 5 times. What is the probability P of getting at least one six?

n = 5

P =

4.ProbabilityHard

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

n = 10p = 0.7

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.9451. Find z to two decimals.

A = 0.9451

z =

6.Normal distributionHard

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. Find the value x below which 90% of them lie (the 90th percentile).

μ = 500σ = 40P = 90%

x = h

7.Confidence intervalsMedium

The blood pressure of 7 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean −2.4 and standard deviation 4.6. Find the margin of error E of the 90% confidence interval for the mean difference.

n = 7d̄ = −2.4s_d = 4.690%t(6) = 1.943

E = mmHg

8.Confidence intervalsHard

A pilot study suggests that about 0.3 of products are defective. How many products n must be inspected so that a 90% confidence interval for the proportion has a margin of error of at most 0.04?

p ≈ 0.3E = 0.0490%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 16 students has a mean score of 81 and a sample standard deviation of 10. Find the test statistic t for H₀: μ = 75.

μ₀ = 75x̄ = 81s = 10n = 16

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test of H₀: μ = 500 against H₁: μ < 500 is done at α = 0.05 with a sample of 25; the standard deviation is known to be 2.5. In reality the mean is 498.6. Find the probability of a Type II error β in percent.

μ₀ = 500μ = 498.6σ = 2.5n = 25α = 0.05

β = %

11.Linear regression & correlationMedium

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

n = 5Σx = 115Σy = 640Σx² = 2735Σy² = 84574Σxy = 15203

r =

12.Linear regression & correlationHard

The table gives 5 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. A new observation (5, 24) is made. Find its residual e = y − ŷ.

n = 5(5, 24)

e =

3Solutions

1.Descriptive statistics

Answer: s = 8.729

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 = 116/5 = 23.2

Squared deviations

Σ(x − x̄)² = (−4.2)² + (−11.2)² + 9.8² + 7.8² + (−2.2)² = 304.8

Standard deviation

s = √(304.8/4) = 8.729

Answer

s = 8.729

2.Descriptive statistics

Answer: z = 1.25

Standardise

z = (180 − 170)/8 = 1.25

Answer

z = 1.25

3.Probability

Answer: P = 0.5981

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.

Complement

P(no six) = (5/6)⁵ = 0.4019

P = 1 − 0.4019 = 0.5981

Answer

P = 0.5981

4.Probability

Answer: P = 0.0001437

Zero and one failure

P(X = 0) = 0.3¹⁰ = 5.905 × 10⁻⁶

P(X = 1) = 10·0.7·0.3⁹ = 0.0001378

Add them

P = 5.905 × 10⁻⁶ + 0.0001378 = 0.0001437

Answer

P = 0.0001437

5.Normal distribution

Answer: z = 1.92

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 + 0.9451)/2 = 0.9726

Read the z-table backwards

Find 0.9726 inside the table and read off z.

z = 1.92

Answer

z = 1.92

6.Normal distribution

Answer: x = 551.3 h

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 = 500 + 1.282·40 = 551.26 h

Answer

x = 551.26 h

7.Confidence intervals

Answer: E = 3.378 mmHg

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.

Critical value

df = 6, t = 1.943

Margin of error

E = 1.943·4.6/√7 = 3.378

Answer

E = 3.378 mmHg

8.Confidence intervals

Answer: n = 356

Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.

Solve for n

n = 1.645²·0.3·0.7/0.04² = 355.2

Round up

n = 356

Answer

n = 356

9.Hypothesis tests & confidence intervals

Answer: t = 2.4

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 = 10/√16 = 10/4 = 2.5

Test statistic

t = (81 − 75)/2.5 = 2.4

Answer

t = 2.4

t_{0.025; df = 15} = 2.131

|t| > 2.131, so H₀ is rejected at the 5% level (two-sided).

10.Hypothesis tests & confidence intervals

Answer: β = 12.4 %

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 = 2.5/√25 = 0.5

x̄_c = 500 − 1.645·0.5 = 499.2

z under the true mean

z = (499.2 − 498.6)/0.5 = 1.155

Type II error

power = P(Z < 1.155) = 87.6%

β = 100% − 87.6% = 12.4%

Answer

β = 12.4%

11.Linear regression & correlation

Answer: r = 0.9883

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 = 15203 − 115·640/5 = 483

S_xx = 2735 − 115²/5 = 90

S_yy = 84574 − 640²/5 = 2654

Correlation coefficient

r = 483/√(90·2654) = 0.9883

Answer

r = 0.9883

12.Linear regression & correlation

Answer: e = 4.8

Means

x̄ = 30/5 = 6

ȳ = 113/5 = 22.6

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 136

Regression line

b = 136/40 = 3.4

a = 22.6 − 3.4·6 = 2.2

ŷ = 2.2 + 3.4·x

Predicted value

ŷ = 2.2 + 3.4·5 = 19.2

Residual

e = 24 − 19.2 = 4.8

Answer

e = 4.8

The point lies above the line.

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