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

A Cronbach's alpha of 0.88 for a 15-item anxiety scale indicates:

AThe scale has a significant effect on anxiety
BScores are stable over six months
CThe scale predicts future anxiety well
DGood internal consistency of the items

2.Research methods in psychologyHard

A study finds that students who sleep more have higher grades. Which design change would best allow a causal conclusion?

AAsk students to report their sleep twice and average the two reports
BMeasure grades more accurately
CRandomly assign students to different sleep schedules
DSurvey more students

3.Descriptive statisticsMedium

Find the sample standard deviation s of the data: 22, 34, 33, 11, 31, 45.

n = 6

s =

4.Descriptive statisticsHard

A quantity is normally distributed with mean 100 and standard deviation 8. What proportion P of values is below 96?

μ = 100σ = 8x = 96

P =

5.ProbabilityMedium

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

5 / 5

P =

6.ProbabilityHard

A test has 6 multiple-choice questions. A student who guesses gets each one right with probability 0.1, independently. What is the probability P of exactly 4 correct answers?

n = 6p = 0.1k = 4

P =

7.Normal distributionMedium

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. What fraction lies between 57 and 67.5 points? Give P as a decimal.

μ = 65σ = 10x₁ = 57x₂ = 67.5

P =

8.Normal distributionHard

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. Find the value x below which 95% of them lie (the 95th percentile).

μ = 32σ = 6P = 95%

x = min

9.Confidence intervalsMedium

The blood pressure of 6 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 9.7 and standard deviation 4.5. Find the margin of error E of the 99% confidence interval for the mean difference.

n = 6d̄ = 9.7s_d = 4.599%t(5) = 4.032

E = mmHg

10.Confidence intervalsHard

The population standard deviation is 26. 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 4?

σ = 26E = 490%

n =

11.Hypothesis tests & confidence intervalsMedium

A sample of 25 students has a mean score of 504.2 and a sample standard deviation of 10. Find the test statistic t for H₀: μ = 500.

μ₀ = 500x̄ = 504.2s = 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 3? The standard deviation is 15. Find the smallest sample size n.

σ = 15E = 399%

n =

13.Linear regression & correlationMedium

The table gives 5 paired observations of x = car age in years and y = price in k€. Find the coefficient of determination r² of the least-squares line.

n = 5

r² =

14.Linear regression & correlationHard

The table gives 6 paired observations of x = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 176?

n = 6ŷ = 176

x =

15.Chi-square testsMedium

The contingency table shows observed counts. For a chi-square test of independence, find the expected count E for the cell “rural, high”.

row total = 78column total = 45N = 135

E =

16.Chi-square testsHard

A chi-square goodness-of-fit test compares observed counts in 5 categories with fully specified expected proportions (no parameters are estimated from the data). 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: D · Good internal consistency of the items

Why this is the answer

Cronbach's alpha estimates internal consistency; values around 0.7 or above are usually considered acceptable. It does not by itself show validity.

Answer

D · Good internal consistency of the items

2.Research methods in psychology

Answer: C · Randomly assign students to different sleep schedules

Why this is the answer

Only manipulating sleep with random allocation controls third variables and allows causal inference, within ethical limits.

Answer

C · Randomly assign students to different sleep schedules

3.Descriptive statistics

Answer: s = 11.6

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 = 176/6 = 29.33

Squared deviations

Σ(x − x̄)² = (−7.333)² + 4.667² + 3.667² + (−18.33)² + 1.667² + 15.67² = 673.3

Standard deviation

s = √(673.3/5) = 11.6

Answer

s = 11.6

4.Descriptive statistics

Answer: P = 0.3085

Standardise

z = −0.5

Standard normal table

Φ(−0.5) = 0.3085

P = 0.3085

Answer

P = 0.3085

5.Probability

Answer: P = 0.2222

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 = 5/10 · 4/9 = 20/90 = 0.2222

Answer

P = 0.2222

6.Probability

Answer: P = 0.001215

Binomial formula

P(X = 4) = 15·0.1⁴·0.9² = 15·0.0001·0.81 = 0.001215

Answer

P = 0.001215

7.Normal distribution

Answer: P = 0.3869

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 = (57 − 65)/10 = −0.8

z₂ = (67.5 − 65)/10 = 0.25

Area between

P = Φ(0.25) − Φ(−0.8) = 0.5987 − 0.2119 = 0.3869

Answer

P = 0.3869

8.Normal distribution

Answer: x = 41.87 min

z for the percentile

Φ(z) = 0.95 → z = 1.645

Back to x

x = 32 + 1.645·6 = 41.87 min

Answer

x = 41.87 min

9.Confidence intervals

Answer: E = 7.407 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 = 5, t = 4.032

Margin of error

E = 4.032·4.5/√6 = 7.407

Answer

E = 7.407 mmHg

10.Confidence intervals

Answer: n = 115

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 = (1.645·26/4)² = 114.3

Round up

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

n = 115

Answer

n = 115

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 = (504.2 − 500)/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 = 166

Solve for n

n = (2.576·15/3)² = 165.9

Round up

n = 166

Answer

n = 166

13.Linear regression & correlation

Answer: r² = 0.9391

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 = 90/5 = 18

Sums of squares and products

S_xx = Σ(x − x̄)² = 10

S_xy = Σ(x − x̄)(y − ȳ) = −36

S_yy = Σ(y − ȳ)² = 138

Coefficient of determination

r² = −36²/(10·138) = 0.9391

Answer

r² = 0.9391

About 93.91% of the variation in y is explained by the line.

14.Linear regression & correlation

Answer: x = 28.57

Means

x̄ = 135/6 = 22.5

ȳ = 817/6 = 136.2

Sums of squares and products

S_xx = Σ(x − x̄)² = 157.5

S_xy = Σ(x − x̄)(y − ȳ) = 1034

Regression line

b = 1034/157.5 = 6.562

a = 136.2 − 6.562·22.5 = −11.48

ŷ = −11.48 + 6.562·x

Solve for x

x = (176 − (−11.48))/6.562 = 28.57

Answer

x = 28.57

15.Chi-square tests

Answer: E = 26

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 count

E = 78·45/135 = 26

Answer

E = 26

16.Chi-square tests

Answer: χ²_crit = 13.28

Degrees of freedom

df = 5 − 1 = 4

Table

χ²_crit(df = 4, α = 0.01) = 13.277

Answer

χ²_crit = 13.277

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

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