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: 23, 26, 38, 31, 26, 57.

n = 6

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 70 and standard deviation 8. What proportion P of values is below 78?

μ = 70σ = 8x = 78

P =

3.ProbabilityMedium

In how many ways N can a committee of 4 people be chosen from a group of 10? The order does not matter.

n = 10k = 4

N =

4.ProbabilityHard

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

n = 9p = 0.4

P =

5.Normal distributionMedium

Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.8212. Find z to two decimals.

A = 0.8212

z =

6.Normal distributionHard

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. Find the value x below which 95% of them lie (the 95th percentile).

μ = 32σ = 6P = 95%

x = min

7.Confidence intervalsMedium

The blood pressure of 13 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean −8 and standard deviation 4.5. Find the lower limit of the 90% confidence interval for the mean difference.

n = 13d̄ = −8s_d = 4.590%t(12) = 1.782

lower limit = mmHg

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 0.5?

σ = 39E = 0.595%

n =

9.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 121.7 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 120.

μ₀ = 120x̄ = 121.7s = 12n = 25

t =

10.Hypothesis tests & confidence intervalsHard

How large must a sample be so that the 90% confidence interval for a mean has a margin of error of at most 2? The standard deviation is 10. Find the smallest sample size n.

σ = 10E = 290%

n =

11.Linear regression & correlationMedium

The table gives 6 paired observations of x = temperature in °C and y = ice creams sold. Find the coefficient of determination r² of the least-squares line.

n = 6

r² =

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 = 59?

n = 5ŷ = 59

x =

3Solutions

1.Descriptive statistics

Answer: s = 12.66

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 = 201/6 = 33.5

Squared deviations

Σ(x − x̄)² = (−10.5)² + (−7.5)² + 4.5² + (−2.5)² + (−7.5)² + 23.5² = 801.5

Standard deviation

s = √(801.5/5) = 12.66

Answer

s = 12.66

2.Descriptive statistics

Answer: P = 0.8413

Standardise

z = 1

Standard normal table

Φ(1) = 0.8413

P = 0.8413

Answer

P = 0.8413

3.Probability

Answer: N = 210

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(10, 4) = (10·9·8·7)/(4·3·2·1) = 210

Answer

N = 210

4.Probability

Answer: P = 0.07054

Zero and one failure

P(X = 0) = 0.6⁹ = 0.01008

P(X = 1) = 9·0.4·0.6⁸ = 0.06047

Add them

P = 0.01008 + 0.06047 = 0.07054

Answer

P = 0.07054

5.Normal distribution

Answer: z = −0.92

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 − 0.8212 = 0.1788

Read the z-table backwards

Find 0.1788 inside the table and read off z.

z = −0.92

Answer

z = −0.92

6.Normal distribution

Answer: x = 41.87 min

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) = 0.95 → z = 1.645

Back to x

x = 32 + 1.645·6 = 41.87 min

Answer

x = 41.87 min

7.Confidence intervals

Answer: lower limit = −10.22 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 = 12, t = 1.782

Margin of error

E = 1.782·4.5/√13 = 2.224

Interval

−8 ± 2.224 → [−10.22, −5.776]

Answer

lower limit = −10.22 mmHg

8.Confidence intervals

Answer: n = 23373

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 = (1.96·39/0.5)² = 23372

Round up

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

n = 23373

Answer

n = 23373

9.Hypothesis tests & confidence intervals

Answer: t = 0.7083

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 = 12/√25 = 12/5 = 2.4

Test statistic

t = (121.7 − 120)/2.4 = 0.7083

Answer

t = 0.7083

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: n = 68

Solve for n

n = (1.645·10/2)² = 67.65

Round up

n = 68

Answer

n = 68

11.Linear regression & correlation

Answer: r² = 0.9722

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 = 126/6 = 21

ȳ = 621/6 = 103.5

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 333

S_yy = Σ(y − ȳ)² = 1630

Coefficient of determination

r² = 333²/(70·1630) = 0.9722

Answer

r² = 0.9722

About 97.22% of the variation in y is explained by the line.

12.Linear regression & correlation

Answer: x = 3.929

Means

x̄ = 25/5 = 5

ȳ = 325/5 = 65

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 224

Regression line

b = 224/40 = 5.6

a = 65 − 5.6·5 = 37

ŷ = 37 + 5.6·x

Solve for x

x = (59 − 37)/5.6 = 3.929

Answer

x = 3.929

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