Sampling and hypothesis testing flashcards

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Population
The whole collection of people or items about which information is wanted.
Census
An investigation in which every member of the population is observed or measured.
Sampling unit
One individual member of the population that could be selected for a sample.
Sampling frame
The list (register, map, database) of sampling units from which the sample is actually drawn.
Statistic
A quantity calculated only from the sample values, containing no unknown parameters (e.g. sample mean, median, largest value).
Sampling distribution
The probability distribution of a statistic over all possible samples of a given size.
Null hypothesis H₀
The claim about a parameter that is assumed true unless the data give enough evidence against it; it contains “=”, e.g. H₀: p = 0.15.
Critical region
The set of values of the test statistic that lead to rejecting H₀. Its boundary value is the critical value.
Actual significance level
P(test statistic lies in the critical region | H₀ true): the probability of rejecting H₀ when it is true.
Binomial probability, mean and variance
P(X = x) = ⁿCₓ pˣ(1 − p)ⁿ⁻ˣ; mean np; variance np(1 − p). (Yes)
Poisson probability, mean and variance
P(X = x) = e^(−λ) λˣ / x!; mean = variance = λ. (Yes)
Upper-tail probability from cumulative tables
P(X ≥ c) = 1 − P(X ≤ c − 1). (No — learn it)
Testing p > 0.5 with the tables
If X ~ B(n, p) then Y = n − X ~ B(n, 1 − p), so P(X ≤ k) = P(Y ≥ n − k). (No — derive it)
Normal approximation to B(n, p)
B(n, p) ≈ N(np, np(1 − p)) for large n with p not far from 0.5; apply a continuity correction. (No — learn it)
Normal approximation to Po(λ)
Po(λ) ≈ N(λ, λ) for large λ (say above 10); apply a continuity correction. (No — learn it)
Continuity correction (upper and lower tails)
P(X ≥ a) ≈ P(Y > a − ½); P(X ≤ a) ≈ P(Y < a + ½). (No — learn it)
Units outside the sampling frame on a population diagram
They can never be chosen, however random the selection, so an incomplete frame can bias the sample.
Bar chart of B(n, p) with a two-tailed critical region
Red bars at both ends; here each tail is the largest set with probability ≤ half the significance level (the “without exceeding” rule).
Normal curve over binomial bars, boundary at 71.5
The bars from 72 upwards start at 71.5, so the area beyond 71.5 approximates P(X ≥ 72).
Standard Normal with the top 5% shaded
One-tailed critical region z ≥ 1.6449; a test statistic beyond it is significant at 5%.
Why is a sample used instead of a census?
It is quicker, cheaper and easier to process, and it is the only option when testing destroys the item.
Disadvantages of a sample
Results vary from sample to sample, and the sample may not be representative (poor frame or non-random selection).
One-tailed or two-tailed?
“Increased” or “decreased” → one-tailed (> or <); “changed” or “different” → two-tailed (≠).
Why is the actual significance level rarely equal to the stated level?
Binomial and Poisson are discrete, so tail probabilities jump. With a “without exceeding” rule it is below the level; with an “as close as possible” rule it can be slightly above.
What does “do not reject H₀” mean?
There is insufficient evidence against H₀ at that level. It does not prove H₀ is true.
Is X̄ − μ a statistic when μ is unknown?
No: it cannot be calculated from the sample values alone.
X ~ B(20, 0.15). Two-tailed critical region at 10%, each tail as close as possible to 5% without exceeding 5%, and actual significance level?
X = 0 (0.0388) or X ≥ 7 (1 − 0.9781 = 0.0219); actual significance level 0.0607.
H₀: p = 0.45, H₁: p < 0.45, n = 30, 8 successes, 5% level.
P(X ≤ 8) = 0.0312 < 0.05: reject H₀; sufficient evidence that p has decreased.
Rate 0.8 per hour; 8 events in 5 hours. Test for an increase at 5%.
X ~ Po(4); P(X ≥ 8) = 1 − 0.9489 = 0.0511 > 0.05: not significant. (CR is X ≥ 9.)
X ~ B(150, 0.4); 72 observed; test H₁: p > 0.4 at 5% with a Normal approximation.
N(60, 36): P(X ≥ 72) ≈ P(Y > 71.5) = P(Z > 1.92) = 1 − 0.9726 = 0.0274 < 0.05: reject H₀.
Po(60) under H₀, 75 observed, H₁: λ larger, 5% level.
N(60, 60): P(Y > 74.5) = P(Z > 1.87) = 1 − 0.9693 = 0.0307 < 0.05: reject H₀.
Values 1, 2, 5 with probabilities 0.6, 0.3, 0.1; sample of 3. Sampling distribution of the median M?
P(M = 1) = 0.648, P(M = 2) = 0.324, P(M = 5) = 0.028.
H₀: p = 0.8, H₁: p < 0.8, n = 10, 5 successes. Probability for the test?
Y = 10 − X ~ B(10, 0.2): P(X ≤ 5) = P(Y ≥ 5) = 1 − 0.9672 = 0.0328.
Writing hypotheses
Use the parameter: H₀: p = 0.45, H₁: p < 0.45 (or λ). Never X, a sample proportion, or words alone.
Justifying a critical value
Quote the probability just inside the level and the one just outside it, e.g. P(X ≥ 12) = 0.0201, P(X ≥ 11) = 0.0426.
Which rule for a critical region?
Use the rule the question states. “As close as possible to 5%”: the nearest tail probability, even if just above 5%. “Without exceeding 5%”: the largest region with probability ≤ 5%.
Two-tailed test at 5%
Compare each tail with 2.5%; add both tails for the actual significance level.
Conclusion wording
Decision plus context, never certain: “sufficient evidence to suggest the rate has increased”.
Poisson over a different period
Scale the mean first: 0.8 per hour for 5 hours is Po(4), not Po(0.8).

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