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 content analysis, inter-rater reliability is checked by:

AUsing a larger sample of materials
BCoding the material twice in one day
CHaving two researchers code the same material and comparing results
DAsking participants afterwards whether the researcher's coding of their material is correct

2.Research methods in psychologyHard

In multiple regression predicting exam scores from study hours and sleep, R² = 0.36. What does this mean?

A36% of students passed the exam
BThe predictors together explain 36% of the variance in exam scores
CStudy hours cause 36% of exam success
DThe correlation between the predictors is 0.36

3.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 35, 21, 31, 44, 54, 48.

n = 6

s =

4.Descriptive statisticsHard

Exam scores are normally distributed with mean 60 and standard deviation 10. Find the z-score z of a score of 48.

μ = 60σ = 10x = 48

z =

5.ProbabilityMedium

In how many ways N can a committee of 2 people be chosen from a group of 11? The order does not matter.

n = 11k = 2

N =

6.ProbabilityHard

A test has 8 multiple-choice questions. A student who guesses gets each one right with probability 0.4, independently. What is the probability P of exactly 5 correct answers?

n = 8p = 0.4k = 5

P =

7.Normal distributionMedium

The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. What fraction lies between 100 and 115.75 points? Give P as a decimal.

μ = 100σ = 15x₁ = 100x₂ = 115.75

P =

8.Normal distributionHard

The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. Drag the cut-off so the shaded bottom 80% is to the left of it. Find the cut-off x.

μ = 500 hσ = 40 h80%

x = h

9.Confidence intervalsMedium

A 99% confidence interval for a mean, based on 50 observations and a known σ, has a margin of error of 5.1. What is the margin of error E₂ if the same study uses 450 observations instead?

99%n₁ = 50E₁ = 5.1n₂ = 450

E₂ =

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 2.5?

σ = 9E = 2.590%

n =

11.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 54.2 and a sample standard deviation of 10. Find the test statistic t for H₀: μ = 50.

μ₀ = 50x̄ = 54.2s = 10n = 25

t =

12.Hypothesis tests & confidence intervalsHard

How large must a sample be so that the 99% confidence interval for a mean has a margin of error of at most 1? The standard deviation is 8. Find the smallest sample size n.

σ = 8E = 199%

n =

13.Linear regression & correlationMedium

The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Their sums are given. Find Pearson's correlation coefficient r from these sums.

n = 5Σx = 120Σy = 382Σx² = 2970Σy² = 30054Σxy = 9435

r =

14.Linear regression & correlationHard

The table gives 5 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. A new observation (3, 13) is made. Find its residual e = y − ŷ.

n = 5(3, 13)

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 = 802 × 3

χ² =

16.Chi-square testsHard

A chi-square test of independence is carried out on a contingency table with 2 rows and 4 columns (not counting the totals). Using the table, find the critical value χ²_crit at significance level α = 0.01.

α = 0.01

χ²_crit =

3Solutions

1.Research methods in psychology

Answer: C · Having two researchers code the same material and comparing results

Why this is the answer

If independent coders produce similar codes for the same data (high agreement or correlation), the coding scheme is reliable.

Answer

C · Having two researchers code the same material and comparing results

2.Research methods in psychology

Answer: B · The predictors together explain 36% of the variance in exam scores

Why this is the answer

R² shows how much of the variability in the outcome is accounted for by all predictors together. It does not establish causation.

Answer

B · The predictors together explain 36% of the variance in exam scores

3.Descriptive statistics

Answer: s = 12.12

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 = 233/6 = 38.83

Squared deviations

Σ(x − x̄)² = (−3.833)² + (−17.83)² + (−7.833)² + 5.167² + 15.17² + 9.167² = 734.8

Standard deviation

s = √(734.8/5) = 12.12

Answer

s = 12.12

4.Descriptive statistics

Answer: z = −1.2

Standardise

z = (48 − 60)/10 = −1.2

Answer

z = −1.2

5.Probability

Answer: N = 55

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(11, 2) = (11·10)/(2·1) = 55

Answer

N = 55

6.Probability

Answer: P = 0.1239

Binomial formula

P(X = 5) = 56·0.4⁵·0.6³ = 56·0.01024·0.216 = 0.1239

Answer

P = 0.1239

7.Normal distribution

Answer: P = 0.3531

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 = (100 − 100)/15 = 0

z₂ = (115.75 − 100)/15 = 1.05

Area between

P = Φ(1.05) − Φ(0) = 0.8531 − 0.5 = 0.3531

Answer

P = 0.3531

8.Normal distribution

Answer: x = 534 h

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) = 0.8

z from the table

z ≈ 0.84

Back to the original units

x = 500 + 0.84·40 = 534 h

Answer

x = 534 h

9.Confidence intervals

Answer: E₂ = 1.7

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 = (2.576/2.576)·√(50/450) = 0.3333

New margin

E₂ = 5.1·0.3333 = 1.7

Answer

E₂ = 1.7

10.Confidence intervals

Answer: n = 36

Solve for n

n = (1.645·9/2.5)² = 35.07

Round up

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

n = 36

Answer

n = 36

11.Hypothesis tests & confidence intervals

Answer: t = 2.1

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 = 10/√25 = 10/5 = 2

Test statistic

t = (54.2 − 50)/2 = 2.1

Answer

t = 2.1

t_{0.025; df = 24} = 2.064

|t| > 2.064, so H₀ is rejected at the 5% level (two-sided).

12.Hypothesis tests & confidence intervals

Answer: n = 425

Solve for n

n = (2.576·8/1)² = 424.7

Round up

n = 425

Answer

n = 425

13.Linear regression & correlation

Answer: r = 0.9546

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 = 9435 − 120·382/5 = 267

S_xx = 2970 − 120²/5 = 90

S_yy = 30054 − 382²/5 = 869.2

Correlation coefficient

r = 267/√(90·869.2) = 0.9546

Answer

r = 0.9546

14.Linear regression & correlation

Answer: e = 3.65

Means

x̄ = 30/5 = 6

ȳ = 82/5 = 16.4

Sums of squares and products

S_xx = Σ(x − x̄)² = 40

S_xy = Σ(x − x̄)(y − ȳ) = 94

Regression line

b = 94/40 = 2.35

a = 16.4 − 2.35·6 = 2.3

ŷ = 2.3 + 2.35·x

Predicted value

ŷ = 2.3 + 2.35·3 = 9.35

Residual

e = 13 − 9.35 = 3.65

Answer

e = 3.65

The point lies above the line.

15.Chi-square tests

Answer: χ² = 5.871

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

urban: 40·20/80 = 10, 40·28/80 = 14, 40·32/80 = 16

rural: 40·20/80 = 10, 40·28/80 = 14, 40·32/80 = 16

Chi-square statistic

χ² = (13 − 10)²/10 + (9 − 14)²/14 + (18 − 16)²/16 + (7 − 10)²/10 + (19 − 14)²/14 + (14 − 16)²/16

χ² = 5.871

Answer

χ² = 5.871

df = 2. At α = 0.05 the critical value is 5.991, so H₀ (independence) is not rejected.

16.Chi-square tests

Answer: χ²_crit = 11.35

Degrees of freedom

df = (2 − 1)·(4 − 1) = 3

Table

χ²_crit(df = 3, α = 0.01) = 11.345

Answer

χ²_crit = 11.345

Reject H₀ if the computed χ² is larger than this value.

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