Research Methods and Statistics for Psychology: cram pack

Formula sheets, the key method steps for every topic and 16 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

Chi-square tests

Test statistic: χ² = Σ(O − E)²/E
Goodness of fit: E = n·p, df = k − 1 − (estimated parameters)
Independence: E = row total × column total / grand total, df = (r − 1)(c − 1)
Critical values (α = 0.05): df 1: 3.841, 2: 5.991, 3: 7.815, 4: 9.488
Decision: reject H₀ if χ² > χ²_crit

2Practice problems

1.Research methods in psychologyMedium

In a field experiment, the researcher:

AOnly observes without manipulating anything
BAnalyses data collected by others
CManipulates the IV in a laboratory
DManipulates the IV in a real-world setting

2.Research methods in psychologyHard

The "replication crisis" in psychology refers to:

AJournals being legally required to reject all replication studies submitted
BCopying of questionnaires without permission
CMany published findings failing to reproduce in new studies
DA shortage of participants for experiments

3.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 45, 16, 16, 41, 38.

n = 5

s =

4.Descriptive statisticsHard

Exam scores are normally distributed with mean 170 and standard deviation 15. Find the z-score z of a score of 140.

μ = 170σ = 15x = 140

z =

5.ProbabilityMedium

A bag holds 3 red and 5 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?

3 / 5

P =

6.ProbabilityHard

A test has 10 multiple-choice questions. A student who guesses gets each one right with probability 0.3, independently. What is the probability P of exactly 1 correct answers?

n = 10p = 0.3k = 1

P =

7.Normal distributionMedium

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. What fraction lies between 49 and 67 points? Give P as a decimal.

μ = 65σ = 10x₁ = 49x₂ = 67

P =

8.Normal distributionHard

The mass X of a product from line 1 and Y from line 2 are independent and normal: X has mean 500 g and standard deviation 8 g, Y has mean 490 g and standard deviation 6 g. Find the probability P that X is greater than Y. Round z to two decimals.

μ_X = 500 g, σ_X = 8 gμ_Y = 490 g, σ_Y = 6 g

P =

9.Confidence intervalsMedium

A sample of 21 items has mean 102 and standard deviation s = 12. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.

n = 21x̄ = 102s = 1299%t(20) = 2.845

E =

10.Confidence intervalsHard

Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 99% confidence interval for the proportion has a margin of error of at most 0.02?

p = 0.5E = 0.0299%

n =

11.Hypothesis tests & confidence intervalsMedium

A sample of 81 students has a mean score of 496.5 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 500.

μ₀ = 500x̄ = 496.5s = 12n = 81

t =

12.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.05 must detect a shift of the mean of 3 units with a power of 80%. The standard deviation is 28. Find the minimum sample size n.

α = 0.05power = 80%δ = 3σ = 28z_α = 1.645z_β = 0.842

n =

13.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 = 15Σy = 305Σx² = 55Σy² = 18991Σxy = 976

r =

14.Linear regression & correlationHard

The table gives 6 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. A new observation (11, 20) is made. Find its residual e = y − ŷ.

n = 6(11, 20)

e =

15.Chi-square testsMedium

The contingency table shows observed counts. Compute the chi-square statistic χ² for the test of independence between the row and column variables. Do not use a continuity correction.

N = 1802 × 2

χ² =

16.Chi-square testsHard

The table shows observed counts. Test at α = 0.05 whether the two variables are independent (no continuity correction). Compute the test statistic χ², then compare it with the critical value from the table.

N = 2003 × 2α = 0.05

χ² =

3Solutions

1.Research methods in psychology

Answer: D · Manipulates the IV in a real-world setting

Why this is the answer

Field experiments, such as staged helping situations in the street, increase ecological validity but give less control over extraneous variables and raise consent issues.

Answer

D · Manipulates the IV in a real-world setting

2.Research methods in psychology

Answer: C · Many published findings failing to reproduce in new studies

Why this is the answer

Large replication projects found many effects were smaller or absent, linked to publication bias, small samples and flexible analysis. Pre-registration is one response.

Answer

C · Many published findings failing to reproduce in new studies

3.Descriptive statistics

Answer: s = 14.1

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 = 156/5 = 31.2

Squared deviations

Σ(x − x̄)² = 13.8² + (−15.2)² + (−15.2)² + 9.8² + 6.8² = 794.8

Standard deviation

s = √(794.8/4) = 14.1

Answer

s = 14.1

4.Descriptive statistics

Answer: z = −2

Standardise

z = (140 − 170)/15 = −2

Answer

z = −2

5.Probability

Answer: P = 0.1071

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.

First and second draw

P = 3/8 · 2/7 = 6/56 = 0.1071

Answer

P = 0.1071

6.Probability

Answer: P = 0.1211

Binomial formula

P(X = 1) = 10·0.3¹·0.7⁹ = 10·0.3·0.04035 = 0.1211

Answer

P = 0.1211

7.Normal distribution

Answer: P = 0.5245

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-scores

z₁ = (49 − 65)/10 = −1.6

z₂ = (67 − 65)/10 = 0.2

Area between

P = Φ(0.2) − Φ(−1.6) = 0.5793 − 0.0548 = 0.5245

Answer

P = 0.5245

8.Normal distribution

Answer: P = 0.8413

Sums and differences of normal variables: if X and Y are independent and normal, X + Y and X − Y are normal too. Means add or subtract: E(X ± Y) = μ_X ± μ_Y. Variances always add, also for a difference: Var(X ± Y) = σ_X² + σ_Y², so σ = √(σ_X² + σ_Y²). Standard deviations never add directly. The total of n independent items with the same distribution has mean n·μ and standard deviation σ·√n. Then standardise as usual: z = (x − mean)/σ and read Φ(z).

Distribution of the difference

μ = 500 − 490 = 10 g, σ = 10 g

Standardise

z = (0 − 10)/10 = −1

Probability

P = 1 − Φ(−1) = 0.8413

Answer

P = 0.8413

9.Confidence intervals

Answer: E = 7.45

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.

Critical value

df = n − 1 = 20, t = 2.845

Margin of error

E = 2.845·12/√21 = 7.45

Answer

E = 7.45

10.Confidence intervals

Answer: n = 4148

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 = 2.576²·0.5·0.5/0.02² = 4147

Round up

n = 4148

Answer

n = 4148

11.Hypothesis tests & confidence intervals

Answer: t = −2.625

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/√81 = 12/9 = 1.333

Test statistic

t = (496.5 − 500)/1.333 = −2.625

Answer

t = −2.625

t_{0.025; df = 80} = 1.99

|t| > 1.99, so H₀ is rejected at the 5% level (two-sided).

12.Hypothesis tests & confidence intervals

Answer: n = 539

Sample size for a test: to detect a true shift δ with power 1 − β in a one-sided z test at level α, the standard error must satisfy δ = (z_α + z_β)·σ/√n, so n = ((z_α + z_β)·σ/δ)², rounded up.

Solve for n

n = ((1.645 + 0.842)·28/3)² = 538.8

Round up

n = 539

Answer

n = 539

13.Linear regression & correlation

Answer: r = 0.9818

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 = 976 − 15·305/5 = 61

S_xx = 55 − 15²/5 = 10

S_yy = 18991 − 305²/5 = 386

Correlation coefficient

r = 61/√(10·386) = 0.9818

Answer

r = 0.9818

14.Linear regression & correlation

Answer: e = 2.29

Means

x̄ = 42/6 = 7

ȳ = 73/6 = 12.17

Sums of squares and products

S_xx = Σ(x − x̄)² = 70

S_xy = Σ(x − x̄)(y − ȳ) = 97

Regression line

b = 97/70 = 1.386

a = 12.17 − 1.386·7 = 2.467

ŷ = 2.467 + 1.386·x

Predicted value

ŷ = 2.467 + 1.386·11 = 17.71

Residual

e = 20 − 17.71 = 2.29

Answer

e = 2.29

The point lies above the line.

15.Chi-square tests

Answer: χ² = 23.33

Chi-square tests: compare observed counts O with the counts E expected under H₀ using χ² = Σ(O − E)²/E. Goodness of fit: E = n·p for each category, df = k − 1 (minus one more for each parameter estimated from the data). Independence in an r × c table: E = row total × column total / grand total, df = (r − 1)(c − 1). Reject H₀ when χ² exceeds the critical value for the chosen α.

Expected counts

new drug: 120·126/180 = 84, 120·54/180 = 36

placebo: 60·126/180 = 42, 60·54/180 = 18

Chi-square statistic

χ² = (70 − 84)²/84 + (50 − 36)²/36 + (56 − 42)²/42 + (4 − 18)²/18

χ² = 23.33

Answer

χ² = 23.33

df = 1. At α = 0.05 the critical value is 3.841, so H₀ (independence) is rejected.

16.Chi-square tests

Answer: χ² = 18.17

Expected counts

plant A: 70, 30

plant B: 42, 18

plant C: 28, 12

Chi-square statistic

χ² = (62 − 70)²/70 + (38 − 30)²/30 + (39 − 42)²/42 + (21 − 18)²/18 + (39 − 28)²/28 + (1 − 12)²/12

χ² = 18.17

Decision

df = 2, χ²_crit = 5.991

18.17 > 5.991

Answer

χ² = 18.17

χ² is above the critical value, so H₀ is rejected: the variables appear to be related.

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