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1Formulas and methods
Evidence-based practice statistics
Sensitivity and specificity
Se = a/(a + c), Sp = d/(b + d)
Predictive values
PPV = a/(a + b), NPV = d/(c + d)
Accuracy
(a + d)/n
Likelihood ratios
LR+ = Se/(1 − Sp), LR− = (1 − Se)/Sp
PPV at prevalence p
Se·p / (Se·p + (1 − Sp)(1 − p))
Odds
odds = p/(1 − p); p = odds/(1 + odds); post-test odds = pre-test odds × LR
Risk measures
RR = EER/CER, ARR = CER − EER, RRR = ARR/CER
Number needed to treat
NNT = 1/ARR, rounded up; NNH = 1/(EER − CER)
Odds ratio
OR = ad/(bc)
Incidence rate
new cases / person-time
Method
Draw the 2 × 2 table and label a, b, c and d before you calculate.
Read sensitivity and specificity down the columns, predictive values along the rows.
For another prevalence, imagine 10 000 people and fill the table with Se, Sp and the prevalence.
For a likelihood ratio, convert the probability to odds, multiply, and convert back.
For a trial, find EER and CER first; then RR = EER/CER, ARR = CER − EER and NNT = 1/ARR rounded up.
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
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.
2Practice problems
1.Evidence-based practice statisticsMedium
Practice exercise (made-up values, not clinical guidance). A case-control study compares 360 patients with a disease (cases) and 720 people without it (controls). The table shows how many in each group were exposed to a risk factor. Find the odds ratio OR for the exposure. Round to two decimal places.
a = 234b = 346c = 126d = 374
OR =
2.Evidence-based practice statisticsHard
Practice exercise (made-up values, not clinical guidance). A patient has a pre-test probability of 15% for a disease. A test with sensitivity 95% and specificity 78% comes back negative. Find the post-test probability p′ of the disease as a percentage. Round to one decimal place.
pre-test 15%Se = 95%Sp = 78%negative
p′ = %
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 28, 25, 32, 25, 14, 33.
n = 6
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 170 and standard deviation 12. Find the z-score z of a score of 164.
μ = 170σ = 12x = 164
z =
5.Confidence intervalsMedium
A sample of 6 items has mean 118.5 and standard deviation s = 15.5. The population is roughly normal. Find the lower limit of the 95% confidence interval for the mean.
n = 6x̄ = 118.5s = 15.595%t(5) = 2.571
lower limit =
6.Confidence intervalsHard
Two fertilisers are tested. Plot group 1: 5 plots, mean yield 100 kg, standard deviation 7 kg. Plot group 2: 7 plots, mean 88 kg, standard deviation 9 kg. Assume equal variances (pooled t, df = n₁ + n₂ − 2). Find the margin of error E of the 90% confidence interval for μ₁ − μ₂.
A sample of 49 students has a mean score of 504.1 and a sample standard deviation of 10. Find the test statistic t for H₀: μ = 500.
μ₀ = 500x̄ = 504.1s = 10n = 49
t =
8.Hypothesis tests & confidence intervalsHard
In a trial, 12 of 80 patients on treatment A recover, and 38 of 150 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 12n₁ = 80x₂ = 38n₂ = 150
z =
3Solutions
1.Evidence-based practice statistics
Answer: OR = 2.01
Likelihood ratios and risk measures: LR+ = sensitivity/(1 − specificity), LR− = (1 − sensitivity)/specificity. In a trial, EER and CER are the event rates with and without treatment: RR = EER/CER, ARR = CER − EER, RRR = ARR/CER and NNT = 1/ARR, rounded up. In a case-control study only the odds ratio OR = ad/(bc) can be used. An incidence rate = new cases/person-time.
Odds of exposure
cases: 234/126 = 1.857
controls: 346/374 = 0.9251
Odds ratio
OR = (234·374)/(346·126) = 2.007 ≈ 2.01
Answer
OR = 2.01
OR above 1: exposure is more common among the cases.
2.Evidence-based practice statistics
Answer: p′ = 1.1 %
Predictive values depend on prevalence: PPV = Se·p/(Se·p + (1 − Sp)(1 − p)) and NPV = Sp(1 − p)/(Sp(1 − p) + (1 − Se)p). With likelihood ratios: pre-test odds = p/(1 − p), post-test odds = pre-test odds × LR, probability = odds/(1 + odds). For a population of N, true positives = N·p·Se and false positives = N·(1 − p)·(1 − Sp). NNT = 1/(CER × RRR), rounded up.
Likelihood ratio
LR− = (1 − 0.95)/0.78 = 0.0641
Pre-test odds
0.15/(1 − 0.15) = 0.1765
Post-test odds
0.1765·0.0641 = 0.01131
Post-test probability
p′ = 0.01131/(1 + 0.01131) = 0.01119 ≈ 1.1%
Answer
p′ = 1.1%
3.Descriptive statistics
Answer: s = 6.853
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.
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 = (164 − 170)/12 = −0.5
Answer
z = −0.5
5.Confidence intervals
Answer: lower limit = 102.2
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 = 5, t = 2.571
Margin of error
E = 2.571·15.5/√6 = 16.27
Interval
118.5 ± 16.27 → [102.2, 134.8]
Answer
lower limit = 102.2
6.Confidence intervals
Answer: E = 8.762 kg
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.
Pooled standard deviation
sp² = (4·7² + 6·9²)/10 = 68.2
sp = 8.258
Margin of error
E = 1.812·8.258·√(1/5 + 1/7) = 8.762
Answer
E = 8.762 kg
7.Hypothesis tests & confidence intervals
Answer: t = 2.87
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/√49 = 10/7 = 1.429
Test statistic
t = (504.1 − 500)/1.429 = 2.87
Answer
t = 2.87
t0.025; df = 48 = 2.011
|t| > 2.011, so H₀ is rejected at the 5% level (two-sided).
8.Hypothesis tests & confidence intervals
Answer: z = −1.81
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 = 12/80 = 0.15
p̂₂ = 38/150 = 0.2533
p̂ = 50/230 = 0.2174
Standard error under H₀
SE = √(0.2174·0.7826·(1/80 + 1/150)) = 0.0571
Test statistic
z = (0.15 − 0.2533)/0.0571 = −1.81
Answer
z = −1.81
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).