Source: Quiz 7 Solutions, Principles of Statistics I, Texas A&M University
Tags: hypothesis testing, z-test, test statistic, p-value, significance level, null hypothesis, alternative hypothesis, reject H0, fail to reject, two-tailed test, one-tailed test, population mean, STAT 211
Hypothesis testing is a structured method for deciding whether sample data provides enough evidence to reject a claim about a population parameter. You set up null and alternative hypotheses, compute a test statistic (here, a z-score), find the p-value, and compare it to the significance level to draw a conclusion. The phrasing of your conclusion matters on exams.
Null hypothesis (H₀)
The default claim about the population parameter, typically a statement of "no difference" or "no effect." It always contains an equality (=, ≤, or ≥).
Alternative hypothesis (H₁ or Hₐ)
The research claim you are trying to find evidence for. It is the complement of H₀ and determines the direction of the test (one-tailed or two-tailed).
Test statistic
A standardised value calculated from the sample data that measures how far the sample result is from the null hypothesis value. For a z-test on a population mean: z* = (x̄ − μ₀) / (σ / √n).
p-value
The probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming H₀ is true. A small p-value means the data is unlikely under H₀.
Significance level (α)
The threshold for rejecting H₀. Common values are 0.05 and 0.01. If the p-value ≤ α, you reject H₀.
Two-tailed test
Used when H₁ contains ≠. The p-value is calculated as 2 × P(Z > |z*|), because extreme values in either direction count as evidence against H₀.
One-tailed test (left or right)
Used when H₁ contains < (left-tailed) or > (right-tailed). The p-value is the area in one tail only.
Fail to reject H₀
The correct phrasing when the p-value > α. You never say "accept H₀," because failing to find evidence against it is not the same as proving it true.
Identify the population parameter in question (here, population mean μ).
The null hypothesis H₀ always states a specific value: H₀: μ = μ₀.
The alternative hypothesis H₁ is determined by the research question:
"Is it different from?" → H₁: μ ≠ μ₀ (two-tailed)
"Is it greater than?" → H₁: μ > μ₀ (right-tailed)
"Is it less than?" → H₁: μ < μ₀ (left-tailed)
The wording of the problem tells you the direction. Look for phrases like "more than," "has decreased," or simply "different from."
The formula:
z* = (x̄ − μ₀) / (σ / √n)
Where:
x̄ = sample mean
μ₀ = hypothesised population mean (from H₀)
σ = population standard deviation (or sample standard deviation s when given)
n = sample size
When σ is unknown but n is small, technically a t-test applies. However, Quiz 7 uses z-test formulas throughout, so follow the formula sheet provided.
Two-tailed (H₁: μ ≠ μ₀): p-value = 2 × P(Z > |z*|)
Right-tailed (H₁: μ > μ₀): p-value = P(Z > z*)
Left-tailed (H₁: μ < μ₀): p-value = P(Z < z*)
Use the standard normal (Z) table or calculator to find these probabilities.
If p-value ≤ α → reject H₀
If p-value > α → fail to reject H₀
The conclusion must be stated in context, using plain language. Two templates:
When you reject H₀: "We reject H₀ at the [α] significance level and conclude that [specific claim in context]."
When you fail to reject H₀: "We do not have sufficient evidence to conclude that [specific claim in context]."
Never say "we accept H₀." Never leave the conclusion in purely statistical language without context.
Z-test statistic for a population mean:
z* = (x̄ − μ₀) / (σ / √n)
Z-test statistic for a population proportion:
z* = (p̂ − p₀) / √(p₀(1 − p₀) / n)
(Proportion test covered in the companion notes.)
Setup: A sample of n = 6 steel beams has mean compressive strength x̄ = 58 psi, standard deviation s = 6 psi. Test whether the true mean differs from 50 psi at α = 0.05.
H₀: μ = 50
H₁: μ ≠ 50
Test statistic:
z* = (58 − 50) / (6 / √6) = 8 / 2.449 ≈ 3.27
p-value: Two-tailed, so p = 2 × P(Z > 3.27) = 2 × 0.011 = 0.022.
Decision: 0.022 < 0.05 → reject H₀.
Conclusion: We reject H₀ at the 0.05 significance level and conclude that the true mean compressive strength of the steel is different from 50 psi.
Setup: The national average daily coffee consumption is μ₀ = 3 cups. A university surveys n = 49 students and finds x̄ = 3.7 cups with s = 1.4. Test whether their students drink more than the national average at α = 0.05.
H₀: μ = 3
H₁: μ > 3
Test statistic:
z* = (3.7 − 3) / (1.4 / √49) = 0.7 / 0.2 = 3.5
p-value: Right-tailed, so p = P(Z > 3.5) = 0.00023.
Decision: 0.00023 < 0.05 → reject H₀.
Conclusion: We reject H₀ at the 0.05 significance level and conclude that the university's students drink more coffee than the average daily consumption of Americans.
Setup: Evening long-distance calls from a city historically average μ₀ = 15.2 minutes. A sample of n = 35 calls has mean x̄ = 14.3 minutes, with σ = 5. Test at α = 0.05 whether the average call length has decreased.
H₀: μ = 15.2
H₁: μ < 15.2
Test statistic:
z* = (14.3 − 15.2) / (5 / √35) = −0.9 / 0.845 ≈ −1.065
p-value: Left-tailed, so p = P(Z < −1.065) = 0.14.
Decision: 0.14 > 0.05 → fail to reject H₀.
Conclusion: We do not have sufficient evidence to conclude that the average evening long-distance call has decreased.
⚠️ The direction of the alternative hypothesis (one-tailed vs. two-tailed) changes how you calculate the p-value. Getting this wrong doubles or halves your p-value.
⚠️ Never say "accept H₀." The correct phrasing is always "fail to reject H₀" or "we do not have sufficient evidence."
⚠️ The conclusion must reference the real-world context, not just the statistical result. Saying "reject H₀" alone is incomplete.
⚠️ For a two-tailed test, multiply the one-tail probability by 2. Forgetting this is one of the most common errors.
⚠️ Check whether the problem gives you σ (population standard deviation) or s (sample standard deviation). Quiz 7 uses the z-test formula regardless, but on later material the distinction matters.
Q: What are the two possible conclusions in hypothesis testing?
A: Reject H₀ (when p-value ≤ α) or fail to reject H₀ (when p-value > α). You never "accept" either hypothesis.
Q: You calculate z = 2.10 for a two-tailed test at α = 0.05. P(Z > 2.10) = 0.0179. What is the p-value, and what do you conclude?*
A: p-value = 2 × 0.0179 = 0.0358. Since 0.0358 < 0.05, reject H₀.
Q: A sample mean is higher than the hypothesised mean, and H₁ is right-tailed. Your z = 1.50 and α = 0.05. P(Z > 1.50) = 0.0668. Do you reject H₀?*
A: No. The p-value of 0.0668 is greater than 0.05, so you fail to reject H₀.
Q: Why do we say "fail to reject H₀" rather than "accept H₀"?
A: Because failing to find sufficient evidence against H₀ does not prove H₀ is true. A different sample or a larger sample might yield a different result.
Q: Write the z-test statistic formula for a population mean from memory.
A: z* = (x̄ − μ₀) / (σ / √n)
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