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
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.
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
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 > χ²crit
2Practice problems
1.Research methods in psychologyMedium
A researcher wants to test whether there is an association between gender (male/female) and preferred film genre (four categories). Which test is appropriate?
- APaired t-test
- BOne-way ANOVA
- CChi-square test of independence
- DPearson's correlation
2.Research methods in psychologyHard
A study of 10 participants finds a large but non-significant effect. What is the most appropriate conclusion?
- AThe study proves the null hypothesis is true, since the effect was not significant
- BThe effect size must be miscalculated
- CThe effect must be a Type I error
- DThe study may have lacked power, so the result is inconclusive
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 11, 55, 57, 51, 54, 20, 43.
n = 7
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 170 and standard deviation 8. What proportion P of values is below 158?
μ = 170σ = 8x = 158
P =
5.ProbabilityMedium
In how many ways N can a committee of 2 people be chosen from a group of 6? The order does not matter.
n = 6k = 2
N =
6.ProbabilityHard
11 components are tested, and each one fails with probability 0.2, independently. What is the probability P that at most one fails?
n = 11p = 0.2
P =
7.Normal distributionMedium
The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. What fraction lies between 460 and 540 h? Give P as a decimal.
μ = 500σ = 40x1 = 460x2 = 540
P =
8.Normal distributionHard
The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. Drag the cut-off so the shaded top 85% is to the right of it. Find the cut-off x.
μ = 100 pointsσ = 15 points85%
x = points
9.Confidence intervalsMedium
A sample of 30 items has mean 81 and standard deviation s = 6. The population is roughly normal. Find the margin of error E of a 90% confidence interval for the mean.
n = 30x̄ = 81s = 690%t(29) = 1.699
E =
10.Confidence intervalsHard
The population standard deviation is 10. How large a sample n do you need so that a 95% confidence interval for the mean has a margin of error of at most 3?
σ = 10E = 395%
n =
11.Hypothesis tests & confidence intervalsMedium
A sample of 49 students has a mean score of 79.5 and a sample standard deviation of 15. Find the test statistic t for H₀: μ = 75.
μ₀ = 75x̄ = 79.5s = 15n = 49
t =
12.Hypothesis tests & confidence intervalsHard
In a trial, 75 of 250 patients on treatment A recover, and 76 of 200 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 75n₁ = 250x₂ = 76n₂ = 200
z =
13.Linear regression & correlationMedium
The table gives 6 paired observations of x = hours studied and y = exam score. Find Pearson's correlation coefficient r.
n = 6
r =
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. Predict ŷ for x = 23.
n = 6x = 23
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 = 1002 × 3
χ² =
16.Chi-square testsHard
A chi-square goodness-of-fit test compares observed counts in 7 categories with fully specified expected proportions (no parameters are estimated from the data). How many degrees of freedom df does the test have?
k = 7
df =
3Solutions
1.Research methods in psychology
Answer: C · Chi-square test of independence
Why this is the answer
The chi-square test compares observed with expected frequencies in a contingency table to test whether two categorical variables are associated.
Answer
C · Chi-square test of independence
2.Research methods in psychology
Answer: D · The study may have lacked power, so the result is inconclusive
Why this is the answer
With small samples, even real effects often fail to reach significance (Type II error). Researchers should report the effect size and confidence interval and replicate with a larger sample.
Answer
D · The study may have lacked power, so the result is inconclusive
3.Descriptive statistics
Answer: s = 18.55
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 = 291/7 = 41.57
Squared deviations
Σ(x − x̄)² = (−30.57)² + 13.43² + 15.43² + 9.429² + 12.43² + (−21.57)² + 1.429² = 2064
Standard deviation
s = √(2064/6) = 18.55
Answer
s = 18.55
4.Descriptive statistics
Answer: P = 0.06681
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 = −1.5
Standard normal table
Φ(−1.5) = 0.06681
P = 0.06681
Answer
P = 0.06681
5.Probability
Answer: N = 15
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(6, 2) = (6·5)/(2·1) = 15
Answer
N = 15
6.Probability
Answer: P = 0.3221
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.
Zero and one failure
P(X = 0) = 0.811 = 0.0859
P(X = 1) = 11·0.2·0.810 = 0.2362
Add them
P = 0.0859 + 0.2362 = 0.3221
Answer
P = 0.3221
7.Normal distribution
Answer: P = 0.6827
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 = (460 − 500)/40 = −1
z₂ = (540 − 500)/40 = 1
Area between
P = Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827
Answer
P = 0.6827
8.Normal distribution
Answer: x = 84.4 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) = 0.15
z from the table
z ≈ −1.04
Back to the original units
x = 100 − 1.04·15 = 84.4 points
Answer
x = 84.4 points
9.Confidence intervals
Answer: E = 1.861
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 = 29, t = 1.699
Margin of error
E = 1.699·6/√30 = 1.861
Answer
E = 1.861
10.Confidence intervals
Answer: n = 43
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 = (1.96·10/3)² = 42.68
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 43
Answer
n = 43
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 = 15/√49 = 15/7 = 2.143
Test statistic
t = (79.5 − 75)/2.143 = 2.1
Answer
t = 2.1
t0.025; df = 48 = 2.011
|t| > 2.011, so H₀ is rejected at the 5% level (two-sided).
12.Hypothesis tests & confidence intervals
Answer: z = −1.786
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 = 75/250 = 0.3
p̂₂ = 76/200 = 0.38
p̂ = 151/450 = 0.3356
Standard error under H₀
SE = √(0.3356·0.6644·(1/250 + 1/200)) = 0.0448
Test statistic
z = (0.3 − 0.38)/0.0448 = −1.786
Answer
z = −1.786
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
13.Linear regression & correlation
Answer: r = 0.9126
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 = 27/6 = 4.5
ȳ = 337/6 = 56.17
Sums of squares and products
Sxx = Σ(x − x̄)² = 17.5
Sxy = Σ(x − x̄)(y − ȳ) = 57.5
Syy = Σ(y − ȳ)² = 226.8
Correlation coefficient
r = 57.5/√(17.5·226.8) = 0.9126
Answer
r = 0.9126
14.Linear regression & correlation
Answer: ŷ = 70.65
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 = 126/6 = 21
ȳ = 385/6 = 64.17
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 227
Regression line
b = 227/70 = 3.243
a = 64.17 − 3.243·21 = −3.933
ŷ = −3.933 + 3.243·x
Prediction
ŷ = −3.933 + 3.243·23 = 70.65
Answer
ŷ = 70.65
15.Chi-square tests
Answer: χ² = 1.528
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: 40·15/100 = 6, 40·40/100 = 16, 40·45/100 = 18
rural: 60·15/100 = 9, 60·40/100 = 24, 60·45/100 = 27
Chi-square statistic
χ² = (8 − 6)²/6 + (14 − 16)²/16 + (18 − 18)²/18 + (7 − 9)²/9 + (26 − 24)²/24 + (27 − 27)²/27
χ² = 1.528
Answer
χ² = 1.528
df = 2. At α = 0.05 the critical value is 5.991, so H₀ (independence) is not rejected.
16.Chi-square tests
Answer: df = 6
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.
Degrees of freedom
df = 7 − 1 = 6
Answer
df = 6
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