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
Introductory Statistics: cram pack
Formula sheets, the key method steps for every topic and 12 practice problems, with full solutions at the back.
1Formulas and methods
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)·pk(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
- Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n, Sxy = Σxy − ΣxΣy/n
- Slope
- b = Sxy/Sxx
- Intercept
- a = ȳ − b·x̄
- Correlation
- r = Sxy/√(SxxSyy)
- Residual
- e = y − ŷ
2Practice problems
1.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 24, 34, 19, 21, 38, 10, 35.
n = 7
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 170 and standard deviation 15. Find the z-score z of a score of 176.
μ = 170σ = 15x = 176
z =
3.ProbabilityMedium
For the distribution shown, find the variance Var(X) of X.
x: 0, 2, 4, 6p(x): 0.05, 0.35, 0.3, 0.3
Var(X) =
4.ProbabilityHard
X has the distribution shown, and Y = −2X + 20. Find the standard deviation σY of Y.
x: 0, 2, 4, 6p(x): 0.35, 0.3, 0.2, 0.15Y = −2X + 20
σY =
5.Normal distributionMedium
Drag z so that the shaded area between −z and z under the standard normal curve is 0.9312. Find z to two decimals.
A = 0.9312
z =
6.Normal distributionHard
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Drag the cut-off so the shaded bottom 5% is to the left of it. Find the cut-off x.
μ = 65 pointsσ = 10 points5%
x = points
7.Confidence intervalsMedium
A 99% confidence interval for a mean, based on 64 observations and a known σ, has a margin of error of 10.2. What is the margin of error E₂ of a 95% interval based on 128 observations?
99%n₁ = 64E₁ = 10.295%n₂ = 128
E₂ =
8.Confidence intervalsHard
The population standard deviation is 25. How large a sample n do you need so that a 90% confidence interval for the mean has a margin of error of at most 1?
σ = 25E = 190%
n =
9.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 20 students, mean score 87.5, standard deviation 8.5. Group 2: 6 students, mean score 97.2, standard deviation 3.5. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).
n₁ = 20x̄₁ = 87.5s₁ = 8.5n₂ = 6x̄₂ = 97.2s₂ = 3.5
t =
10.Hypothesis tests & confidence intervalsHard
In a trial, 40 of 100 patients on treatment A recover, and 51 of 150 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 40n₁ = 100x₂ = 51n₂ = 150
z =
11.Linear regression & correlationMedium
The table gives 6 paired observations of x = car age in years and y = price in k€. Their sums are given. Find Pearson's correlation coefficient r from these sums.
n = 6Σx = 21Σy = 108Σx² = 91Σy² = 2080Σxy = 330
r =
12.Linear regression & correlationHard
The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 4.5.
n = 5x = 4.5
ŷ =
3Solutions
1.Descriptive statistics
Answer: s = 10.19
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 = 181/7 = 25.86
Squared deviations
Σ(x − x̄)² = (−1.857)² + 8.143² + (−6.857)² + (−4.857)² + 12.14² + (−15.86)² + 9.143² = 622.9
Standard deviation
s = √(622.9/6) = 10.19
Answer
s = 10.19
2.Descriptive statistics
Answer: z = 0.4
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.
Standardise
z = (176 − 170)/15 = 0.4
Answer
z = 0.4
3.Probability
Answer: Var(X) = 3.31
Discrete random variables: a probability distribution lists every value x a variable can take with its probability p(x); the probabilities are between 0 and 1 and add up to 1. The expected value (mean) is E(X) = Σx·p(x), the long-run average. The variance is Var(X) = Σ(x − μ)²·p(x) = E(X²) − μ², and the standard deviation σ = √Var(X). For a linear change Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X) and σY = |a|·σX; adding a constant does not change the spread.
Mean
μ = 0·0.05 + 2·0.35 + 4·0.3 + 6·0.3 = 3.7
Mean of the squares
E(X²) = 0·0.05 + 4·0.35 + 16·0.3 + 36·0.3 = 17
Variance
Var(X) = 17 − 3.7² = 3.31
Answer
Var(X) = 3.31
4.Probability
Answer: σY = 4.247
Discrete random variables: a probability distribution lists every value x a variable can take with its probability p(x); the probabilities are between 0 and 1 and add up to 1. The expected value (mean) is E(X) = Σx·p(x), the long-run average. The variance is Var(X) = Σ(x − μ)²·p(x) = E(X²) − μ², and the standard deviation σ = √Var(X). For a linear change Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X) and σY = |a|·σX; adding a constant does not change the spread.
Spread of X
μ = 2.3, Var(X) = 4.51, σX = 2.124
Linear change
σY = |−2|·2.124 = 4.247
Answer
σY = 4.247
The mean does move: E(Y) = −2·2.3 + 20 = 15.4.
5.Normal distribution
Answer: z = 1.82
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 + 0.9312)/2 = 0.9656
Read the z-table backwards
Find 0.9656 inside the table and read off z.
z = 1.82
Answer
z = 1.82
6.Normal distribution
Answer: x = 48.6 points
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
Φ(z) = 0.05
z from the table
z ≈ −1.64
Back to the original units
x = 65 − 1.64·10 = 48.6 points
Answer
x = 48.6 points
7.Confidence intervals
Answer: E₂ = 5.488
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.
Ratio of margins
E₂/E₁ = (1.96/2.576)·√(64/128) = 0.538
New margin
E₂ = 10.2·0.538 = 5.488
Answer
E₂ = 5.488
8.Confidence intervals
Answer: n = 1692
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 = (1.645·25/1)² = 1691
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 1692
Answer
n = 1692
9.Hypothesis tests & confidence intervals
Answer: t = −2.696
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.
Pooled variance
sp² = (19·8.5² + 5·3.5²)/24 = 59.75
sp = 7.73
Standard error
SE = 7.73·√(1/20 + 1/6) = 3.598
Test statistic
t = (87.5 − 97.2)/3.598 = −2.696
Answer
t = −2.696
With df = 24, the two-sided 5% critical value is 2.064; |t| is larger, so H₀ is rejected.
10.Hypothesis tests & confidence intervals
Answer: z = 0.9659
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.
Sample proportions
p̂₁ = 40/100 = 0.4
p̂₂ = 51/150 = 0.34
p̂ = 91/250 = 0.364
Standard error under H₀
SE = √(0.364·0.636·(1/100 + 1/150)) = 0.06212
Test statistic
z = (0.4 − 0.34)/0.06212 = 0.9659
Answer
z = 0.9659
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
11.Linear regression & correlation
Answer: r = −0.9839
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
Sxy = 330 − 21·108/6 = −48
Sxx = 91 − 21²/6 = 17.5
Syy = 2080 − 108²/6 = 136
Correlation coefficient
r = −48/√(17.5·136) = −0.9839
Answer
r = −0.9839
12.Linear regression & correlation
Answer: ŷ = 31.45
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 = 15/5 = 3
ȳ = 140/5 = 28
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 23
Regression line
b = 23/10 = 2.3
a = 28 − 2.3·3 = 21.1
ŷ = 21.1 + 2.3·x
Prediction
ŷ = 21.1 + 2.3·4.5 = 31.45
Answer
ŷ = 31.45
ExamProblems · examproblems.com/courses/statistics-1