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: 10, 36, 27, 34, 21.

n = 5

s =

2.Descriptive statisticsHard

A quantity is normally distributed with mean 100 and standard deviation 15. What proportion P of values is below 85?

μ = 100σ = 15x = 85

P =

3.ProbabilityMedium

A fair die is rolled 4 times. What is the probability P of getting at least one six?

n = 4

P =

4.ProbabilityHard

A test has 9 multiple-choice questions. A student who guesses gets each one right with probability 0.7, independently. What is the probability P of exactly 1 correct answers?

n = 9p = 0.7k = 1

P =

5.Normal distributionMedium

Drag z so that the shaded area between −z and z under the standard normal curve is 0.8029. Find z to two decimals.

A = 0.8029

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 90% of them lie (the 90th percentile).

μ = 65σ = 10P = 90%

x = points

7.Confidence intervalsMedium

The blood pressure of 16 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 6.3 and standard deviation 0.6. Find the margin of error E of the 99% confidence interval for the mean difference.

n = 16d̄ = 6.3s_d = 0.699%t(15) = 2.947

E = mmHg

8.Confidence intervalsHard

The population standard deviation is 18. 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 1?

σ = 18E = 199%

n =

9.Hypothesis tests & confidence intervalsMedium

10 employees take a typing test before and after a training course. For each person the difference d = after − before (words per minute) is recorded. The mean difference is 9.33 and the standard deviation of the differences is 11.8. Find the paired test statistic t for H₀: μ_d = 0.

n = 10d̄ = 9.33s_d = 11.8

t =

10.Hypothesis tests & confidence intervalsHard

A one-sided z test of H₀: μ = 80 against H₁: μ > 80 is done at α = 0.10 with a sample of 64; the standard deviation is known to be 4. In reality the mean is 80.7. Find the power 1 − β of the test in percent.

μ₀ = 80μ = 80.7σ = 4n = 64α = 0.10

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 = 367Σx² = 135Σy² = 27201Σxy = 1884

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. For which value x does the line predict ŷ = 84?

n = 6ŷ = 84

x =

3Solutions

1.Descriptive statistics

Answer: s = 10.55

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 = 128/5 = 25.6

Squared deviations

Σ(x − x̄)² = (−15.6)² + 10.4² + 1.4² + 8.4² + (−4.6)² = 445.2

Standard deviation

s = √(445.2/4) = 10.55

Answer

s = 10.55

2.Descriptive statistics

Answer: P = 0.1587

Standardise

z = −1

Standard normal table

Φ(−1) = 0.1587

P = 0.1587

Answer

P = 0.1587

3.Probability

Answer: P = 0.5177

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)⁴ = 0.4823

P = 1 − 0.4823 = 0.5177

Answer

P = 0.5177

4.Probability

Answer: P = 0.0004133

Binomial formula

P(X = 1) = 9·0.7¹·0.3⁸ = 9·0.7·6.561 × 10⁻⁵ = 0.0004133

Answer

P = 0.0004133

5.Normal distribution

Answer: z = 1.29

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 + 0.8029)/2 = 0.9015

Read the z-table backwards

Find 0.9015 inside the table and read off z.

z = 1.29

Answer

z = 1.29

6.Normal distribution

Answer: x = 77.82 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) = 0.9 → z = 1.282

Back to x

x = 65 + 1.282·10 = 77.82 points

Answer

x = 77.82 points

7.Confidence intervals

Answer: E = 0.442 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 = 15, t = 2.947

Margin of error

E = 2.947·0.6/√16 = 0.442

Answer

E = 0.442 mmHg

8.Confidence intervals

Answer: n = 2150

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 = (2.576·18/1)² = 2150

Round up

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

n = 2150

Answer

n = 2150

9.Hypothesis tests & confidence intervals

Answer: t = 2.5

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 of the mean difference

SE = 11.8/√10 = 3.731

Test statistic

t = 9.33/3.731 = 2.5

Answer

t = 2.5

With df = 9, the two-sided 5% critical value is 2.262; |t| is larger, so H₀ is rejected.

10.Hypothesis tests & confidence intervals

Answer: 1 − β = 54.7 %

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 = 4/√64 = 0.5

x̄_c = 80 + 1.282·0.5 = 80.64

z under the true mean

z = (80.64 − 80.7)/0.5 = −0.118

Power

power = P(Z > −0.118) = 54.7%

1 − β = 54.7%

Answer

1 − β = 54.7%

11.Linear regression & correlation

Answer: r = 0.9551

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 = 1884 − 25·367/5 = 49

S_xx = 135 − 25²/5 = 10

S_yy = 27201 − 367²/5 = 263.2

Correlation coefficient

r = 49/√(10·263.2) = 0.9551

Answer

r = 0.9551

12.Linear regression & correlation

Answer: x = 5.688

Means

x̄ = 33/6 = 5.5

ȳ = 498/6 = 83

Sums of squares and products

S_xx = Σ(x − x̄)² = 17.5

S_xy = Σ(x − x̄)(y − ȳ) = 93

Regression line

b = 93/17.5 = 5.314

a = 83 − 5.314·5.5 = 53.77

ŷ = 53.77 + 5.314·x

Solve for x

x = (84 − 53.77)/5.314 = 5.688

Answer

x = 5.688

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