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

Experts check whether the items of a maths exam cover all topics in the syllabus. This assesses:

AContent validity
BTest-retest reliability
CEcological validity
DPredictive validity

2.Research methods in psychologyHard

A meta-analysis combines 40 studies of a therapy. Why might its overall effect size still be too large?

AMeta-analyses always give smaller effects than the single studies they include
BMeta-analyses cannot include randomised trials
CCombining studies always removes all bias
DPublication bias means unpublished null studies may be missing

3.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 47, 43, 21, 26, 53.

n = 5

s =

4.Descriptive statisticsHard

A quantity is normally distributed with mean 60 and standard deviation 5. What proportion P of values is above 52.5?

μ = 60σ = 5x = 52.5

P =

5.ProbabilityMedium

In how many ways N can a committee of 2 people be chosen from a group of 14? The order does not matter.

n = 14k = 2

N =

6.ProbabilityHard

A test has 12 multiple-choice questions. A student who guesses gets each one right with probability 0.3, independently. What is the probability P of exactly 5 correct answers?

n = 12p = 0.3k = 5

P =

7.Normal distributionMedium

The bag weights are normally distributed with mean 250 g and standard deviation 5 g. What fraction lies between 246 and 254.25 g? Give P as a decimal.

μ = 250σ = 5x₁ = 246x₂ = 254.25

P =

8.Normal distributionHard

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Find the value x below which 25% of them lie (the 25th percentile).

μ = 65σ = 10P = 25%

x = points

9.Confidence intervalsMedium

A sample of 8 items has mean 12 and standard deviation s = 2.5. The population is roughly normal. Find the upper limit of the 95% confidence interval for the mean.

n = 8x̄ = 12s = 2.595%t(7) = 2.365

upper limit =

10.Confidence intervalsHard

The population standard deviation is 9. 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 2?

σ = 9E = 290%

n =

11.Hypothesis tests & confidence intervalsMedium

Two production lines make the same part. Line 1: a sample of 50 parts has mean length 113.5 mm, and the line's known standard deviation is 5 mm. Line 2: a sample of 30 parts has mean 118.7 mm, known standard deviation 9 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).

n₁ = 50x̄₁ = 113.5 mmσ₁ = 5 mmn₂ = 30x̄₂ = 118.7 mmσ₂ = 9 mm

z =

12.Hypothesis tests & confidence intervalsHard

A one-sided z test at α = 0.05 must detect a shift of the mean of 4 units with a power of 90%. The standard deviation is 14. Find the minimum sample size n.

α = 0.05power = 90%δ = 4σ = 14z_α = 1.645z_β = 1.282

n =

13.Linear regression & correlationMedium

The table gives 5 paired observations of x = car age in years and y = price in k€. Find Pearson's correlation coefficient r.

n = 5

r =

14.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. Predict ŷ for x = 6.5.

n = 5x = 6.5

ŷ =

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

A chi-square goodness-of-fit test checks whether counts in 5 categories follow a Poisson distribution whose mean is estimated from the same data. How many degrees of freedom df does the test have?

k = 5

df =

3Solutions

1.Research methods in psychology

Answer: A · Content validity

Why this is the answer

Content validity is the extent to which a measure covers all relevant aspects of the construct, usually judged by experts.

Answer

A · Content validity

2.Research methods in psychology

Answer: D · Publication bias means unpublished null studies may be missing

Why this is the answer

If null results are less often published, the pooled effect is inflated. Funnel plots and searches for unpublished data help detect and reduce this.

Answer

D · Publication bias means unpublished null studies may be missing

3.Descriptive statistics

Answer: s = 13.82

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 = 190/5 = 38

Squared deviations

Σ(x − x̄)² = 9² + 5² + (−17)² + (−12)² + 15² = 764

Standard deviation

s = √(764/4) = 13.82

Answer

s = 13.82

4.Descriptive statistics

Answer: P = 0.9332

Standardise

z = −1.5

Standard normal table

Φ(−1.5) = 0.06681

P = 1 − 0.06681 = 0.9332

Answer

P = 0.9332

5.Probability

Answer: N = 91

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(14, 2) = (14·13)/(2·1) = 91

Answer

N = 91

6.Probability

Answer: P = 0.1585

Binomial formula

P(X = 5) = 792·0.3⁵·0.7⁷ = 792·0.00243·0.08235 = 0.1585

Answer

P = 0.1585

7.Normal distribution

Answer: P = 0.5905

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 = (246 − 250)/5 = −0.8

z₂ = (254.25 − 250)/5 = 0.85

Area between

P = Φ(0.85) − Φ(−0.8) = 0.8023 − 0.2119 = 0.5905

Answer

P = 0.5905

8.Normal distribution

Answer: x = 58.26 points

z for the percentile

Φ(z) = 0.25 → z = −0.6745

Back to x

x = 65 + −0.6745·10 = 58.26 points

Answer

x = 58.26 points

9.Confidence intervals

Answer: upper limit = 14.09

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 = 7, t = 2.365

Margin of error

E = 2.365·2.5/√8 = 2.09

Interval

12 ± 2.09 → [9.91, 14.09]

Answer

upper limit = 14.09

10.Confidence intervals

Answer: n = 55

Solve for n

n = (1.645·9/2)² = 54.8

Round up

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

n = 55

Answer

n = 55

11.Hypothesis tests & confidence intervals

Answer: z = −2.907

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.

Standard error of the difference

SE = √(5²/50 + 9²/30) = √3.2 = 1.789

Test statistic

z = (113.5 − 118.7)/1.789 = −2.907

Answer

z = −2.907

|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).

12.Hypothesis tests & confidence intervals

Answer: n = 105

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 + 1.282)·14/4)² = 104.9

Round up

n = 105

Answer

n = 105

13.Linear regression & correlation

Answer: r = −0.9862

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 = 76/5 = 15.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = −31

S_yy = Σ(y − ȳ)² = 98.8

Correlation coefficient

r = −31/√(10·98.8) = −0.9862

Answer

r = −0.9862

14.Linear regression & correlation

Answer: ŷ = 87.1

Means

x̄ = 25/5 = 5

ȳ = 404/5 = 80.8

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = 42

Regression line

b = 42/10 = 4.2

a = 80.8 − 4.2·5 = 59.8

ŷ = 59.8 + 4.2·x

Prediction

ŷ = 59.8 + 4.2·6.5 = 87.1

Answer

ŷ = 87.1

15.Chi-square tests

Answer: χ² = 31.76

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

men: 120·117/180 = 78, 120·63/180 = 42

women: 60·117/180 = 39, 60·63/180 = 21

Chi-square statistic

χ² = (61 − 78)²/78 + (59 − 42)²/42 + (56 − 39)²/39 + (4 − 21)²/21

χ² = 31.76

Answer

χ² = 31.76

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

16.Chi-square tests

Answer: df = 3

Degrees of freedom

df = 5 − 1 − 1 = 3

Answer

df = 3

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