A researcher knows the population standard deviation is σ = 10. She takes a sample of n = 25 and observes x̄ = 104. She wants to test H₀: μ = 100. What is the correct z-statistic?
Az = (104 − 100) / 10 = 0.4
Bz = (104 − 100) / (10/√25) = 2.0
Cz = (104 − 100) / √(10/25) = 6.3
Dz = (104 − 100) / (10 × 25) = 0.016
The test statistic divides by the standard error σ/√n = 10/√25 = 10/5 = 2, giving z = 4/2 = 2.0. Option A uses σ alone (=10) instead of σ/√n — the most common error. Dividing by σ ignores the effect of sample size: a sample mean of 104 from n=25 observations is far more surprising than a single observation of 104, because averaging reduces spread. The standard error σ/√n is what characterizes the spread of sample means.
Question 2 Multiple Choice
After computing z = 2.1 and a two-tailed p-value of 0.036, a student writes: 'There is a 3.6% chance that the sample mean equals the null value.' What is wrong with this statement?
ANothing — the p-value gives the probability that x̄ equals μ₀
BThe p-value is the probability of observing a z-statistic this extreme or more extreme, assuming H₀ is true — not the probability that x̄ equals a specific value
CThe p-value gives the probability that H₀ is true, not a probability about x̄
DNothing — the statement is equivalent to saying the result is statistically significant at α = 0.05
The p-value is P(|Z| ≥ 2.1 | H₀ true) — a probability about the test statistic under the assumption that H₀ holds. It says nothing directly about whether x̄ equals μ₀ (a specific value has probability 0 for a continuous distribution) or about whether H₀ is true. Misinterpreting p as the probability that H₀ is true (option C) is an equally common error. The correct interpretation is: 'If H₀ were true, we would see a result this extreme only 3.6% of the time by chance.'
Question 3 True / False
The standard error σ/√n is smaller than σ because averaging over more observations reduces the variability of the sample mean.
TTrue
FFalse
Answer: True
Each individual observation has variance σ². The sample mean x̄ averages n independent observations, so its variance is σ²/n and its standard deviation is σ/√n. As n increases, the standard error shrinks — large samples produce sample means clustered tightly around μ. This is why a sample mean of 104 is far more statistically surprising (stronger evidence against H₀: μ=100) when n=100 than when n=4, even though x̄ is the same in both cases.
Question 4 True / False
A z-test is appropriate whenever the sample size is large (n > 30), even when the population standard deviation σ is unknown.
TTrue
FFalse
Answer: False
The z-test requires knowing σ, the population standard deviation — not just the sample standard deviation s. When σ is unknown (which is almost always in practice), substituting s for σ changes the distribution of the test statistic from standard normal to a t-distribution with n−1 degrees of freedom. The t-distribution has heavier tails to account for the extra uncertainty from estimating σ from data. For large n, the t-distribution approximates the normal, but the correct test is still technically a t-test, not a z-test.
Question 5 Short Answer
Explain why the z-test formula uses σ/√n in the denominator rather than σ, and what σ/√n represents.
Think about your answer, then reveal below.
Model answer: σ/√n is the standard error of the sample mean — the standard deviation of the sampling distribution of x̄. Individual observations vary with standard deviation σ, but the test is about x̄, not a single observation. By the central limit theorem, x̄ is approximately normally distributed with mean μ and standard deviation σ/√n. Dividing by σ/√n standardizes x̄ into z-score units of 'how many standard errors is x̄ from μ₀?' Using σ instead of σ/√n would ignore sample size entirely: a sample mean of 104 from n=1 and from n=10,000 would produce the same z, which is wrong — larger samples provide far stronger evidence.
The standard error is the key quantity that links sample size to inferential power. It embodies the core logic: more data → tighter sampling distribution of x̄ → more surprising a given deviation from μ₀ → larger |z| → smaller p-value.