Sections 13.1 – 13.4
The Logic and Language of Hypothesis Testing
Why Testing Exists · Claims · Null & Alternative Hypotheses · One- and Two-Sided Tests
Section 13.1
Why Hypothesis Testing Exists
Suppose someone tells you that a coin is fair — that it lands heads half the time in the long run. You flip it 20 times and get 13 heads. Should you believe the coin is fair? Maybe. Getting 13 heads in 20 flips is unusual but not impossible. What if you got 18 heads out of 20? That would be far more surprising. At some point, the data become so inconsistent with the claim that you begin to doubt it. But where exactly is that point? When does a result go from "unusual but believable" to "too extreme to accept"?
Hypothesis testing exists because sample results vary from sample to sample due to sampling variability. A single sample result that differs from a claim could simply reflect normal sampling variation, or it could reflect a genuinely false claim. There needs to be a principled, consistent method for deciding which interpretation is more supported by the data.
The Seven-Step Hypothesis Testing Process
- State the null and alternative hypotheses.
- Choose the significance level α.
- Check the conditions for the test.
- Calculate the test statistic.
- Find the p-value.
- Make a decision: reject or fail to reject H₀.
- Interpret the conclusion in context.
The Logic of Hypothesis Testing
Assume the claim is true. Ask: if it were true, how likely would we be to see a sample result at least as extreme as the one we observed? If that probability is very low, the data are unlikely under the assumption — and we have reason to doubt the claim. If that probability is not low, the data are consistent with the assumption.
Section 13.2
Claims, Evidence, and Uncertainty
Not every claim is a statistical hypothesis. A statistical hypothesis must be a specific, testable claim about a population parameter — a number that describes the entire population. The evidence used to evaluate a claim comes from sample data. The key question is always: how far is the observed statistic from the claimed value, measured in units of standard error?
| Testable Statistical Claims | Not Directly Testable |
|---|
| The proportion of voters who support a policy is 0.50. | "This drug is effective." (Too vague — effective compared to what?) |
| The mean wait time at a clinic is 30 minutes. | "Students today are less motivated." (Motivation undefined numerically.) |
| The proportion of defective items is less than 0.02. | "The economy is doing well." (No specific parameter stated.) |
Uncertainty Is Always Present
Even a very extreme test statistic does not prove that the claim is false with absolute certainty. Hypothesis testing does not eliminate uncertainty — it quantifies it. Conclusions are probabilistic, not certain. The goal is to make the best decision possible, given the evidence.
Section 13.3
Null and Alternative Hypotheses
The Two Competing Hypotheses
H₀ (Null Hypothesis) — The default assumption. States that there is no effect, no difference, or that a specific claimed parameter value is correct. Always a statement of equality. Assumed true until the data provide strong evidence against it.
Hₐ (Alternative Hypothesis) — The competing claim. States that the parameter differs from the null value — larger, smaller, or simply not equal. Always a statement of inequality. Represents what the researcher is trying to find evidence for.
Writing Hypotheses in Notation
| Parameter | Null Hypothesis | Two-Sided Hₐ | Upper-Tailed Hₐ | Lower-Tailed Hₐ |
|---|
Proportion p | H₀: p = p₀ | Hₐ: p ≠ p₀ | Hₐ: p > p₀ | Hₐ: p < p₀ |
Mean μ | H₀: μ = μ₀ | Hₐ: μ ≠ μ₀ | Hₐ: μ > μ₀ | Hₐ: μ < μ₀ |
Example 13.3.1
Writing hypotheses for a proportion
- ProblemA city council claims that 70% of residents support a new transit plan. A researcher suspects the true proportion is different from 70%.
- ThinkThe council's claim is p = 0.70. The researcher suspects a difference in either direction — no prediction about which way. This calls for a two-sided alternative.
- RepresentLet p = the population proportion of residents who support the transit plan.
- Hypotheses
H₀: p = 0.70
Hₐ: p ≠ 0.70
- InterpretThe null hypothesis assumes the council's claim is correct. The alternative says the true proportion differs from 70% in some direction.
Example 13.3.2
Writing hypotheses for a mean
- ProblemA nutrition label states a serving of cereal contains 200 calories on average. A consumer advocacy group believes the actual count is higher than labeled.
- ThinkThe labeled value is μ₀ = 200. The advocacy group suspects the true mean is higher — one-sided, upper-tailed.
- RepresentLet μ = the true mean number of calories per serving.
- Hypotheses
H₀: μ = 200
Hₐ: μ > 200
- InterpretThe null hypothesis assumes the label is correct. The alternative says the true mean calorie count exceeds 200.
Section 13.4
One-Sided and Two-Sided Tests
| Test Type | Alternative Hypothesis | When to Use |
|---|
| Two-sided | Hₐ: p ≠ p₀ | Research question asks whether the parameter differs from the claimed value, without expecting a direction. |
| Upper-tailed (one-sided) | Hₐ: p > p₀ | Prior context or theory specifically predicts the parameter is larger than the claimed value. |
| Lower-tailed (one-sided) | Hₐ: p < p₀ | Prior context or theory specifically predicts the parameter is smaller than the claimed value. |
Data Ethics — Choosing After Seeing the Data
The direction of the alternative hypothesis must be determined before the data are analyzed, based on the research question. Choosing the direction after seeing the data — so that the result appears more significant — is a form of statistical misconduct called p-hacking or data dredging. It inflates false positive rates and undermines the integrity of the test.
For each situation, write H₀ and Hₐ in notation and words. Identify the test as one-sided or two-sided.
- A candy company claims its bags contain exactly 500 g of candy on average. A consumer group believes the bags are underfilled.
- A standard coin is tested to determine whether the probability of heads equals 0.50.
- A school counselor believes more than 30% of students experience significant test anxiety.
- A quality manager thinks the proportion of correctly filled boxes may have changed from the claimed 0.99.
- A researcher wants to determine whether mean sleep duration differs from 7 hours per night.
Sections 13.5 – 13.9
Test Statistics, p-Values, and Decisions
Test Statistics · p-Values · Significance Levels · Making Decisions · Interpreting Results
Section 13.5
Test Statistics as Measures of Evidence
The General Form of a Test Statistic
test statistic = (observed statistic − null hypothesis value) / standard error
This ratio answers: if the null hypothesis were true, how many standard errors away from the expected value is the sample statistic? A test statistic close to 0 means the data are consistent with H₀. A test statistic far from 0 means the data are surprising under H₀.
The z-Test Statistic for a Proportion
Under the null hypothesis that p = p₀, the sampling distribution of p̂ is approximately normal (when conditions are met). The test statistic uses p₀ in the standard error — not p̂ — because the entire test is conducted under the assumption that the null hypothesis is true.
z-Test Statistic for a Proportion
z = (p̂ − p₀) / √[p₀(1 − p₀) / n]
where p̂ is the sample proportion, p₀ is the null hypothesis value, and n is the sample size.
The t-Test Statistic for a Mean
t-Test Statistic for a Mean
t = (x̄ − μ₀) / (s / √n)
where x̄ is the sample mean, μ₀ is the null hypothesis value, s is the sample standard deviation, and n is the sample size. The degrees of freedom equal n − 1.
Example 13.5.1
Calculating a z-test statistic for a proportion
- ProblemA school district claims 60% of students meet the reading proficiency standard (p₀ = 0.60). A sample of n = 120 students finds 64 meet the standard. Calculate the z-test statistic.
- ThinkThe sample proportion is p̂ = 64/120. The null value is p₀ = 0.60. Use the z-test formula with p₀ in the denominator.
- Calculate
p̂ = 64 / 120 ≈ 0.533
Standard error = √(0.60 × 0.40 / 120)
= √(0.24 / 120)
= √0.002
≈ 0.0447
z = (0.533 − 0.60) / 0.0447
= −0.067 / 0.0447
≈ −1.50
- CheckThe sample proportion 0.533 is lower than 0.60, so the negative sign makes sense. A z-score of −1.50 is about one and a half standard errors below the null value — noticeable but not extreme.
- InterpretThe sample proportion of 53.3% is approximately 1.50 standard errors below the null hypothesis value of 60%. This is the starting point for evaluating how surprising the data are.
Test statistic: z ≈ −1.50
Section 13.6
p-Values
Definition of the p-Value
The p-value is the probability, calculated assuming the null hypothesis is true, of observing a result at least as extreme as the one actually observed in the sample. A small p-value means the observed result would be very unlikely if H₀ were true — this is evidence against H₀. A large p-value means the result is not unusual under H₀.
Calculating the p-Value by Test Type
| Alternative Hypothesis | Test Type | p-Value Calculation |
|---|
Hₐ: p ≠ p₀ | Two-sided | p-value = 2 × P(Z > |z|) |
Hₐ: p > p₀ | Upper-tailed | p-value = P(Z > z) |
Hₐ: p < p₀ | Lower-tailed | p-value = P(Z < z) |
Common Misinterpretation of the p-Value
A p-value of 0.03 does not mean "there is a 3% chance the null hypothesis is true." It means "if the null hypothesis were true, there would be only about a 3% chance of observing a result at least as extreme as this one." The null hypothesis is either true or false — it is not a random event with a probability.
Example 13.6.1
Calculating and interpreting a p-value
- ProblemIn Example 13.5.1, the test statistic was z = −1.50 for a two-sided test of whether p = 0.60. Find and interpret the p-value.
- ThinkTwo-sided test. The p-value is 2 × P(Z < −1.50). Both tails are included because we are testing for a difference in either direction.
- Calculate
P(Z < −1.50) ≈ 0.0668
p-value = 2 × 0.0668 = 0.1336
- CheckA p-value of 0.134 is not small. Under the assumption that 60% of students meet the standard, a sample proportion this far from 0.60 (in either direction) would occur about 13.4% of the time by chance alone.
- InterpretIf the district's claim that p = 0.60 were true, there would be approximately a 13.4% chance of observing a sample proportion as far from 0.60 as the one in this study. This is not strong evidence against the null hypothesis.
p-value ≈ 0.134
Graph Instruction
Draw a standard normal curve centered at zero. Mark the test statistic z = −1.50 on the left side. Shade the area to the left of −1.50. Because this is a two-sided test, also shade the mirror area to the right of +1.50. The total shaded area equals the p-value of approximately 0.134.
Section 13.7
Significance Levels
The p-value measures how surprising the data are under the null hypothesis. But how surprising is surprising enough? To make a formal decision, researchers use a significance level, written α (the Greek letter alpha). The significance level is the threshold probability below which the p-value must fall for the result to be considered statistically significant — that is, for the null hypothesis to be rejected.
The significance level must be chosen before the data are collected and analyzed, based on the context and the consequences of making an error.
| Significance Level | Common Use | Interpretation |
|---|
α = 0.05 | Social sciences, biology, medicine | Willing to accept a 5% chance of incorrectly rejecting H₀. |
α = 0.01 | Clinical trials, quality control | Stricter standard; used when false positives would be costly. |
α = 0.10 | Exploratory or preliminary research | More lenient; appropriate when Type II errors are costly. |
The Decision Rule
If p-value ≤ α: Reject H₀. The data provide statistically significant evidence in favor of the alternative hypothesis.
If p-value > α: Fail to reject H₀. The data do not provide sufficient evidence to reject the null hypothesis.
Note: The conclusion is "fail to reject H₀," never "accept H₀" or "prove H₀." The null hypothesis is never proven — only retained as a reasonable assumption in the absence of sufficient evidence.
Section 13.8
Making a Decision
Template for Stating a Conclusion
When H₀ is rejected:
"At the α significance level, there is sufficient evidence to conclude that [the alternative hypothesis claim in plain language]."
When H₀ is not rejected:
"At the α significance level, there is not sufficient evidence to conclude that [the alternative hypothesis claim in plain language]."
Example 13.8.1
Making a decision and stating a conclusion
- ProblemUsing Examples 13.5.1 and 13.6.1: does the data provide evidence that the district's claim of 60% proficiency is incorrect? Use α = 0.05.
- ThinkThe p-value was approximately 0.134. Compare to α = 0.05.
- Decision
0.134 > 0.05 → Fail to reject H₀
- InterpretAt the 5% significance level, there is not sufficient evidence to conclude that the proportion of students meeting the reading proficiency standard differs from 60%. The sample result of 53.3% is consistent with sampling variability from a population where 60% of students meet the standard.
- NoteThis does not mean the district's claim is definitely correct. It means the data collected do not provide strong enough evidence to dispute it.
Conclusion: Fail to reject H₀. Insufficient evidence against the 60% claim.
Section 13.9
Interpreting Results in Context
Every conclusion in a hypothesis test must be tied to the original question, the variable being measured, the population being studied, and the specific claim being evaluated. Vague conclusions — "we rejected the null hypothesis" — are not useful. Return to the original scenario and use its language. A good conclusion answers: What was being measured? What was the claim? What did the data show? Was the evidence strong enough?
What a Test Does Not Tell You
A hypothesis test does not identify why a result occurred, prove causation, guarantee that the alternative hypothesis is true, confirm exact population values, or replace judgment and further research. A test result should be interpreted alongside the research design, data quality, sample size, potential biases, and practical context.
For each scenario, state the decision and write a complete conclusion in context.
- A test produces z = 2.10 for a two-sided test. Find the p-value using the standard normal table. At α = 0.05, state the decision.
- H₀: p = 0.30, Hₐ: p > 0.30, α = 0.05, p-value = 0.031. The study asked whether more than 30% of shoppers prefer a new product design.
- H₀: μ = 80, Hₐ: μ ≠ 80, α = 0.01, p-value = 0.042. The study asked whether mean exam scores differ from 80 points.
- A student says the p-value of 0.08 means "there is an 8% chance H₀ is true." Explain the error and state the correct interpretation.
- H₀: p = 0.10, Hₐ: p < 0.10, α = 0.05, p-value = 0.089. The study asked whether fewer than 10% of manufactured parts are defective.
Sections 13.10 – 13.14
Errors, Full Tests, and Practical Significance
Type I & II Errors · z-Test for Proportions · t-Test for Means · Significance vs. Importance · Common Mistakes
Section 13.10
Type I and Type II Errors
Any decision based on data can be wrong. In hypothesis testing, there are exactly two ways a decision can be incorrect. A Type I Error occurs when the null hypothesis is rejected even though it is actually true — a false positive, concluding an effect exists when it does not. A Type II Error occurs when the null hypothesis is not rejected even though it is actually false — a false negative, failing to detect an effect that actually exists.
Type I Error
(False Positive)
Probability = α |
Correct Decision
Probability = 1 − α |
Correct Decision (Power)
Probability = 1 − β |
Type II Error
(False Negative)
Probability = β |
Power = 1 − β
The power of a test is the probability of correctly rejecting the null hypothesis when it is false. A powerful test is good at detecting real effects. Increasing sample size, increasing α, or detecting larger effects all increase power. Reducing α (making the test stricter) decreases power and increases the risk of Type II error.
Example 13.10.1
Identifying error types in context
- ProblemA pharmaceutical company tests whether a new vaccine reduces flu infection rates. H₀: the vaccine has no effect. Describe Type I and Type II errors in this context.
- Type I ErrorConcluding the vaccine is effective when it actually has no effect. This could lead to widespread distribution of a useless vaccine, wasting public health resources and exposing patients to potential side effects without benefit.
- Type II ErrorConcluding there is insufficient evidence the vaccine works, when it actually is effective. This means a genuinely effective vaccine is not deployed, leading to preventable illness.
- InterpretIn public health contexts, both errors have serious consequences. The appropriate significance level and study design should reflect careful consideration of both risks, the cost of the vaccine, and the severity of the illness being prevented.
Section 13.11
Tests for Proportions
Conditions for the One-Sample z-Test for a Proportion
1. Random sample: Data come from a random sample or randomized experiment.
2. Normality (large sample): Both np₀ ≥ 10 and n(1 − p₀) ≥ 10. Ensures the sampling distribution of p̂ is approximately normal.
3. Independence: Observations are independent. If sampling without replacement, the sample size should be less than 10% of the population.
Example 13.11.1
Full z-test for a proportion — seven steps
- ProblemA national study reported that 45% of high school students get fewer than eight hours of sleep on school nights. A local high school surveys a random sample of 200 students; 98 report fewer than eight hours. Does this provide evidence the local school's proportion differs from the national rate? Use α = 0.05.
- Hypotheses
Let p = the true proportion of students at this high school
who get fewer than 8 hours of sleep.
H₀: p = 0.45
Hₐ: p ≠ 0.45 (two-sided)
- Conditions
Random sample: ✓ stated in the problem.
Normality: np₀ = 200 × 0.45 = 90 ≥ 10; n(1 − p₀) = 200 × 0.55 = 110 ≥ 10. ✓
Independence: Assume school has > 2,000 students. ✓
- Calculate
p̂ = 98 / 200 = 0.49
SE = √(0.45 × 0.55 / 200) = √(0.2475 / 200) = √0.0012375 ≈ 0.03518
z = (0.49 − 0.45) / 0.03518 = 0.04 / 0.03518 ≈ 1.14
- p-Value
Two-sided: p-value = 2 × P(Z > 1.14)
P(Z > 1.14) ≈ 0.1271
p-value = 2 × 0.1271 = 0.2542
- DecisionSince 0.2542 > 0.05, fail to reject H₀.
- ConcludeAt the 5% significance level, there is not sufficient evidence to conclude that the proportion of students at this high school who get fewer than eight hours of sleep differs from the national rate of 45%. The sample result of 49% is consistent with normal sampling variation from a population where 45% of students are sleep-deprived.
z ≈ 1.14 | p-value ≈ 0.254 | Fail to reject H₀
Section 13.12
Tests for Means
The one-sample t-test for a mean is used when the variable is quantitative, the population standard deviation is unknown (nearly always the case in practice), and the goal is to test a claim about μ. The test statistic follows a t-distribution with n − 1 degrees of freedom. The t-distribution is symmetric and bell-shaped, like the normal distribution, but with heavier tails. As sample size increases, the t-distribution approaches the normal distribution.
Conditions for the One-Sample t-Test for a Mean
1. Random sample: Data from a random sample or randomized experiment.
2. Normality: The population is approximately normal, or n ≥ 30 so the Central Limit Theorem applies. For smaller samples, check for strong skewness or outliers.
3. Independence: Observations independent; sample size < 10% of population if sampling without replacement.
Example 13.12.1
Full t-test for a mean — seven steps
- ProblemA school claims students spend an average of 45 minutes on homework per night. A researcher believes the actual average is higher. A random sample of 25 students shows x̄ = 52 minutes and s = 14 minutes. Test the researcher's claim at α = 0.05.
- Hypotheses
Let μ = the true mean homework time per night for students at this school.
H₀: μ = 45
Hₐ: μ > 45 (one-sided, upper-tailed)
- Conditions
Random sample: ✓ stated.
Normality: n = 25 is small; assume approximately normal distribution (no strong skewness or outliers stated). ✓
Independence: Assume school has > 250 students. ✓
- Calculate
t = (x̄ − μ₀) / (s / √n)
= (52 − 45) / (14 / √25)
= 7 / (14 / 5)
= 7 / 2.8
= 2.50
Degrees of freedom: df = n − 1 = 24
- p-ValueUpper-tailed test: p-value = P(T > 2.50) with df = 24. From a t-table or technology: p-value ≈ 0.010.
- DecisionSince 0.010 < 0.05, reject H₀.
- ConcludeAt the 5% significance level, there is sufficient evidence to conclude that the mean homework time for students at this school is greater than 45 minutes per night. The sample mean of 52 minutes is significantly higher than the claimed 45 minutes, and this difference is unlikely to be due to chance alone.
t = 2.50 | df = 24 | p-value ≈ 0.010 | Reject H₀
Technology Note — Finding t-Distribution p-Values
To find the p-value for a t-test using spreadsheet software, use T.DIST.RT(t, df) for an upper-tailed test, T.DIST(t, df, TRUE) for a lower-tailed test, and 2 × T.DIST.RT(|t|, df) for a two-sided test. On a graphing calculator, use the t-distribution area function with the appropriate degrees of freedom and tail direction.
Section 13.13
Statistical Significance versus Practical Importance
A result is statistically significant when the p-value is less than or equal to α. Statistical significance means the data are unlikely under the null hypothesis — not that the effect is large, meaningful, or important. This distinction is critical, and it is frequently misunderstood in practice, in media reports, and even in published research.
When the sample size is very large, even trivially small differences from the null hypothesis can produce very small p-values. Practical importance (also called effect size or practical significance) refers to whether the magnitude of the observed effect is large enough to matter in the real world. This requires judgment, domain knowledge, and contextual understanding — not just calculation.
Example 13.13.1
Statistical versus practical significance
- SituationA technology company tests a new website interface with n = 10,000 users. Average time on site increases from 18.0 minutes to 18.4 minutes. The hypothesis test produces p-value = 0.002.
- StatisticalYes — statistically significant at α = 0.05. The increase of 0.4 minutes is unlikely to be due to chance alone with this large a sample.
- PracticalWhether 24 seconds is a meaningful increase depends on the business context. 10,000 users × 0.4 minutes = 4,000 extra minutes of engagement daily. This might be commercially significant — or trivial compared to the cost of the redesign. The data alone cannot answer this question.
- ConcludeStatistical significance confirms the effect is real. Practical importance requires additional judgment about whether the size of the effect justifies action.
Data Ethics — Reporting Significance Responsibly
Reporting a result as "significant" without clarifying that the effect may not be practically important can be misleading. Statistical significance does not mean the effect is large. Responsible communication of statistical results always includes context about effect size and practical meaning, not just p-values.
Section 13.14
Common Hypothesis Testing Mistakes
Mistake 1 — Confusing Fail to Reject with Accept
"Since the p-value was 0.08, we accept the null hypothesis."
Correct: "Fail to reject H₀" means the data did not provide sufficient evidence against H₀ — not that H₀ is proven true. The null hypothesis could still be false; the sample may simply not have been large or varied enough to detect the difference.
Mistake 2 — Misinterpreting the p-Value
"p-value = 0.04 means there is a 4% chance H₀ is true."
Correct: The p-value is the probability of observing data at least this extreme given that H₀ is true — not the probability that H₀ is true. The null hypothesis is either true or false; its truth value is not a probability.
Mistake 3 — Choosing the Alternative Hypothesis After Seeing the Data
Noticing that p̂ = 0.47 > 0.40 and then writing Hₐ: p > 0.40 to make the one-sided p-value appear smaller.
Correct: The direction of Hₐ must be determined before the data are analyzed, based on the research question. Choosing it after seeing the data is p-hacking and inflates false positive rates.
Mistake 4 — Ignoring Conditions
Using a z-test for a proportion when np₀ = 50 × 0.01 = 0.5, which is far below 10.
Correct: Check all conditions before proceeding. If the normality condition fails, the normal approximation is invalid and a different method (such as an exact binomial test) should be used.
Mistake 5 — Equating Statistical Significance with Practical Importance
"The p-value was 0.001, so the effect is very large and definitely meaningful."
Correct: A small p-value with a large sample size can reflect a trivially small effect. Always report the size of the effect alongside the p-value and significance decision.
Mistake 6 — Treating One Test as Definitive
"We ran one test and got p = 0.04, so the matter is settled."
Correct: A single hypothesis test is not the final word on a question. Replication — repeating a study with different samples — is essential. Results can depend on the specific sample, the chosen α, and data collection conditions.
Mistake 7 — Ignoring Type II Error
"We failed to reject H₀, so there's probably no real effect."
Correct: A large p-value in a small study may mean the sample was too small to detect a real effect. Low statistical power means failure to reject H₀ is weak evidence for H₀. Always consider sample size and power.
Mistake 8 — Multiple Testing Without Adjustment
Testing 20 hypotheses at α = 0.05 and reporting all significant results without adjustment.
Correct: With 20 tests, one significant result is expected by chance alone even if all null hypotheses are true. When many tests are conducted simultaneously, the significance threshold should be adjusted (e.g., using a Bonferroni correction), or results should be interpreted with caution.
Conduct full seven-step hypothesis tests. Identify errors where indicated.
- A city health department claims 25% of adults smoke regularly. In a random sample of 300 adults, 66 say they smoke. Conduct a full two-sided z-test at α = 0.05.
- A coffee shop advertises a mean caffeine content of 200 mg per large coffee. A researcher samples 36 coffees: x̄ = 214 mg, s = 30 mg. Conduct a full two-sided t-test at α = 0.05.
- A school administrator claims 85% of students graduate within four years. A researcher samples 200 graduates; 158 graduated within four years. Conduct a lower-tailed z-test at α = 0.01.
- A student sets up Hₐ: p > 0.40 after seeing that p̂ = 0.47. What is wrong? What should the student have done instead?
- A study with n = 50,000 produces p-value = 0.0003. The observed difference is 0.2 points on a 100-point scale. Is this practically important? Explain what additional information you would need.
Chapter 13 — Back Matter
Chapter Summary
- Sec 13.1Hypothesis testing is a structured, seven-step process for evaluating statistical claims. It begins with a default assumption, measures how far data deviate from it, and decides whether the deviation is large enough to be convincing.
- Sec 13.2Statistical hypotheses must be testable claims about population parameters. Evidence comes from sample data measured in units of standard error. Uncertainty is always present; conclusions are probabilistic, not certain.
- Sec 13.3H₀ is the default assumption of no effect or a specific parameter value. Hₐ is the competing claim. H₀ is always a statement of equality; Hₐ is always an inequality. Both are written in terms of population parameters before data are collected.
- Sec 13.4Two-sided tests ask whether the parameter differs in either direction. One-sided tests specify a direction. The choice must be made before seeing the data. Choosing afterward (p-hacking) is statistical misconduct.
- Sec 13.5A test statistic measures how many standard errors the observed statistic is from the null value. For proportions, use a z-statistic with p₀ in the standard error. For means, use a t-statistic with df = n − 1.
- Sec 13.6The p-value is the probability of observing a result at least as extreme as the one obtained, assuming H₀ is true. It is not the probability that H₀ is true. Small p-values are evidence against H₀; large p-values are not evidence for H₀.
- Sec 13.7The significance level α is the threshold for making a formal decision. If p-value ≤ α, reject H₀. If p-value > α, fail to reject H₀. α must be chosen before data collection based on context and consequences.
- Sec 13.8Conclusions must be stated in plain language in context. Rejection means sufficient evidence for Hₐ. Failure to reject means insufficient evidence against H₀ — not proof that H₀ is true.
- Sec 13.9Every conclusion must reference the original question, the variable, the population, and the claim. A hypothesis test does not prove causation, explain why a result occurred, or guarantee the alternative hypothesis is true.
- Sec 13.10Type I error: rejecting a true H₀ (probability = α). Type II error: failing to reject a false H₀ (probability = β). Power = 1 − β. These errors trade off against each other and have different consequences in different contexts.
- Sec 13.11The one-sample z-test for a proportion requires random sampling, both np₀ ≥ 10 and n(1 − p₀) ≥ 10, and independence. The z-statistic uses p₀ in the standard error formula.
- Sec 13.12The one-sample t-test for a mean uses t = (x̄ − μ₀)/(s/√n) with df = n − 1. It requires random sampling, approximate normality or n ≥ 30, and independence. The t-distribution has heavier tails than the normal.
- Sec 13.13Statistical significance (p-value ≤ α) is not the same as practical importance. Large samples can detect trivially small effects. Practical importance requires judgment about effect size, real-world meaning, and context.
- Sec 13.14Eight recurring mistakes: confusing fail-to-reject with acceptance; misinterpreting p-values; choosing Hₐ after data; ignoring conditions; equating significance with importance; treating one test as definitive; ignoring Type II error; multiple testing without adjustment.
Key Terms Review
null hypothesis
The default assumption that there is no effect or that the parameter equals the claimed value. Written H₀. Always a statement of equality.
alternative hypothesis
The competing claim that the parameter differs from the null hypothesis value. Written Hₐ. Always a statement of inequality.
test statistic
A standardized measure of how far the sample result is from the null hypothesis value, expressed in units of standard error.
p-value
The probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. Not the probability that H₀ is true.
significance level
The threshold probability α below which the p-value must fall to reject H₀. Must be chosen before data collection. Common values: 0.01, 0.05, 0.10.
one-sided test
A hypothesis test in which Hₐ specifies a direction — greater than or less than the null value. Uses one tail of the distribution.
two-sided test
A hypothesis test in which Hₐ specifies only that the parameter differs from the null value, without specifying direction. Uses both tails.
Type I error
Rejecting the null hypothesis when it is actually true. A false positive. The probability of a Type I error equals α.
Type II error
Failing to reject the null hypothesis when it is actually false. A false negative. The probability of a Type II error is β.
power
The probability of correctly rejecting the null hypothesis when it is false. Power = 1 − β. Increases with sample size and effect size.
z-test for a proportion
A hypothesis test for a population proportion using the normal distribution. Requires np₀ ≥ 10, n(1 − p₀) ≥ 10, and a random sample.
t-test for a mean
A hypothesis test for a population mean using the t-distribution when σ is unknown. Degrees of freedom = n − 1.
statistical significance
The conclusion that the p-value is small enough (below α) to reject the null hypothesis. Does not imply the effect is large or practically meaningful.
practical importance
A judgment about whether the size of the observed effect is large enough to matter in context. Requires domain knowledge and contextual reasoning beyond p-values.
p-hacking
The practice of choosing the direction of the alternative hypothesis or selecting analyses after seeing the data to achieve a significant result. Undermines the validity of hypothesis tests.
fail to reject
The conclusion when the p-value exceeds α. Means the data do not provide sufficient evidence against H₀. Does not mean H₀ is proven true.
Concept Review Questions
- Explain the difference between the null hypothesis and the alternative hypothesis. Which one is assumed true at the start of a test? Which one represents what the researcher hopes to find evidence for?
- What is the significance level α, and when should it be chosen? What does it represent in terms of error probability?
- Explain the difference between failing to reject H₀ and accepting H₀. Why does this distinction matter?
- A student says, "Our p-value was 0.03, so there is a 3% chance the null hypothesis is true." Explain what is wrong with this statement and provide the correct interpretation.
- Describe a situation in which a Type I error would be more serious than a Type II error. Then describe a situation in which a Type II error would be more serious than a Type I error.
- Why can statistical significance not be equated with practical importance? Give an example in which a result is statistically significant but practically trivial.
- A researcher conducts a study and fails to reject H₀. A colleague says, "So the null hypothesis is true." What should the researcher say in response?
- Why is it important to check conditions before conducting a z-test for a proportion?
Proportion and Mean Test Practice
For each problem, follow all seven steps: state hypotheses, choose α, check conditions, calculate the test statistic, find the p-value, make a decision, and write a conclusion in context.
- A political candidate claims she has the support of more than 50% of likely voters. Her campaign surveys 400 likely voters and finds 214 support her. Conduct a full hypothesis test at α = 0.05.
- A bottling company claims its bottles contain a mean of 500 mL of water. A quality inspector samples 36 bottles and finds x̄ = 496 mL, s = 12 mL. Conduct a two-sided t-test at α = 0.05.
- A quality control analyst checks whether the defect rate for a new production line is less than 5%. In a sample of 500 units, 21 are defective. Conduct a full hypothesis test at α = 0.05.
- A researcher believes more than 40% of teenagers in a region spend more than four hours per day on screens. A sample of 150 teenagers shows 70 spend more than four hours. Conduct a full test at α = 0.10.
- A fast-food chain claims drive-through wait time averages 3 minutes. A consumer group samples 50 visits and finds x̄ = 3.4 min, s = 1.2 min. Conduct a two-sided t-test at α = 0.05.
Error Analysis
Each problem contains an error in reasoning or calculation. Identify it and provide the correct approach.
Problem 1
At α = 0.05, p-value = 0.08: "We accept the null hypothesis."
What is the error? What is the correct conclusion?
Problem 2
z = −2.20 for a one-sided upper-tailed test; student finds p-value = P(Z > 2.20)
Is this correct? For an upper-tailed test with z = −2.20, how should the p-value be calculated?
Problem 3
After seeing p̂ = 0.47 > 0.40, student sets Hₐ: p > 0.40 to get a one-sided p-value
What is wrong with this approach and what are its consequences?
Problem 4
"p-value = 0.001, so the effect is very large and definitely meaningful."
Explain the error. What additional information is needed to assess practical importance?
Problem 5
z-test for a proportion used with p₀ = 0.01, n = 50: np₀ = 0.5
What condition fails? What should the student do instead of using the z-test?
Chapter 13 Checkpoint
Chapter 13 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 13
Checkpoint Problem 1 — Hypothesis Setup and z-Test
A local library claims that 40% of its cardholders visited the library at least once in the past month. A survey of 180 randomly selected cardholders finds that 63 visited at least once.
- Define the population, the variable, and the parameter of interest.
- Write H₀ and Hₐ in notation. Use a two-sided test.
- Check all conditions for the z-test for a proportion.
- Calculate the test statistic z. Show all steps.
- Find the p-value using a standard normal table or technology.
- At α = 0.05, state your decision and write a full conclusion in context.
- State whether a Type I or Type II error could have occurred. Describe what each would mean in this context and explain which might be more consequential.
Checkpoint Problem 2 — t-Test for a Mean
A bottling company claims that bottles contain a mean of 500 mL of water. A quality inspector samples 36 bottles and finds a mean of 496 mL with a standard deviation of 12 mL.
- Write H₀ and Hₐ for a two-sided test.
- Check the conditions for the t-test for a mean.
- Calculate the t-test statistic and identify the degrees of freedom.
- Find the approximate p-value using technology or a t-table.
- At α = 0.05, state your decision and write a complete conclusion in context.
Checkpoint Problem 3 — Interpreting a Published Result
A published health report states: "A new sleep supplement was tested in a sample of 5,000 adults. Those taking the supplement slept an average of 7.05 hours, compared to the control group mean of 7.00 hours. The result was statistically significant (p = 0.03)."
- What null and alternative hypotheses are implied by this study?
- At α = 0.05, is the result statistically significant? Explain.
- Is the result practically important? What factors did you consider?
- What additional information would help you evaluate whether this result matters in a real-world context?
Checkpoint Problem 4 — Errors and Consequences
A food safety inspector tests whether the proportion of food samples from a supplier that fail safety standards exceeds 0.02. The inspector uses α = 0.05.
- Write H₀ and Hₐ.
- Describe what a Type I error means in this context and what its consequences might be.
- Describe what a Type II error means in this context and what its consequences might be.
- Should the inspector use a small or large significance level, given the public health stakes? Explain your reasoning.
Looking Ahead
Chapter 14: Statistical Modeling and Data-Based Decisions
Chapter 13 gave you the tools to evaluate evidence under uncertainty — to write and test claims, calculate and interpret p-values, make formal decisions, and recognize limits of significance. Chapter 14 closes the book by bringing together the full statistical investigation process: asking questions, collecting data, displaying data, summarizing variation, modeling relationships, using probability, estimating, testing, and communicating conclusions.
Chapter 14 also addresses data ethics, responsible interpretation, model limitations, and capstone project design. It prepares you to apply everything you have learned in this book to real investigations — and to carry this reasoning into college statistics, data science, economics, public policy, science, medicine, and any field where evidence matters.
The logic of hypothesis testing extends further in future courses. College statistics introduces tests for two proportions, two means, chi-square tests for categorical data, ANOVA for multiple means, and regression hypothesis tests. AP Statistics and college mathematical statistics derive these methods more formally. Data science courses apply them to A/B testing, machine learning evaluation, and algorithmic fairness. Every extension uses the same seven-step logic you learned here.
MGU Library Connection
- Hypothesis Testing Reference — Appendix N of this textbook
- Normal Distribution Reference — Appendix K of this textbook (z-score areas for p-value calculations)
- Sampling Distribution Reference — Appendix L of this textbook (standard error and sampling variability)
- Confidence Interval Reference — Appendix M of this textbook (estimation and inference connections)
- Calculator and Spreadsheet Skills — Appendix O (T.DIST functions, z-test and t-test procedures)
- Statistical Writing Guide — Appendix P (writing conclusions, reporting results responsibly)
- MGU Data Science Foundations — A/B Testing and Experimental Evaluation pages in the MGU Library
- MGU Dictionary entries: null hypothesis, alternative hypothesis, p-value, significance level, Type I error, Type II error, power, t-distribution