The Central Idea
A hypothesis test compares two statements about a population parameter.
The first is the null hypothesis, written
\[ H_0. \]The second is the alternative hypothesis, often written
\[ H_A \]or
\[ H_1. \]The test asks whether the observed data are sufficiently incompatible with the null hypothesis to favor the alternative.
The Null Hypothesis
The null hypothesis is the reference claim used to construct the test's probability model.
It often represents:
- no difference;
- no effect;
- no association;
- a specified benchmark value;
- the status quo;
- a model parameter equal to a particular value.
The Alternative Hypothesis
The alternative hypothesis describes the departure from the null that the study is designed to detect.
Depending on the question, it may say that a parameter is:
- different from a benchmark;
- greater than a benchmark;
- less than a benchmark;
- different between two populations;
- associated with another variable.
The Hypotheses Must Refer to Population Quantities
Hypotheses are statements about population parameters or probability models, not about observed sample statistics.
For a population mean,
\[ H_0:\mu=50 \]is a valid null hypothesis.
Writing
\[ H_0:\bar{x}=50 \]is usually inappropriate because \(\bar{x}\) is already determined by the observed sample.
Equality Belongs in the Null Hypothesis
In standard point-null hypothesis testing, the null contains the equality condition.
For example,
\[ H_0:\mu=100 \]versus
\[ H_A:\mu\neq100. \]The equality gives the precise parameter value used to generate the null reference distribution.
Why Equality Is Needed
The test statistic requires a specific null value.
For a mean,
\[ t = \frac{ \bar{x}-\mu_0 }{ s/\sqrt{n} }. \]The quantity
\[ \mu_0 \]comes from the equality specified by the null hypothesis.
Two-Sided Alternatives
A two-sided alternative asks whether the parameter differs from the null value in either direction.
The standard form is
\[ \boxed{ H_0:\theta=\theta_0 } \]versus
\[ \boxed{ H_A:\theta\neq\theta_0 }. \]When to Use a Two-Sided Alternative
Use a two-sided alternative when departures in either direction matter.
For example, if a manufacturing process is supposed to have mean fill volume
\[ 500\text{ mL}, \]then both underfilling and overfilling may be important.
A natural test is
\[ H_0:\mu=500 \]versus
\[ H_A:\mu\neq500. \]Right-Sided Alternatives
A right-sided or right-tailed alternative asks whether a parameter is greater than the null value.
Its form is
\[ \boxed{ H_0:\theta=\theta_0 } \]versus
\[ \boxed{ H_A:\theta>\theta_0 }. \]When to Use a Right-Sided Alternative
Use a right-sided alternative when only an increase is relevant to the research question.
For example, suppose a new process is intended to increase average output above
\[ 75 \]units per hour.
Then
\[ H_0:\mu=75 \]versus
\[ H_A:\mu>75. \]Left-Sided Alternatives
A left-sided or left-tailed alternative asks whether a parameter is less than the null value.
Its form is
\[ \boxed{ H_0:\theta=\theta_0 } \]versus
\[ \boxed{ H_A:\theta<\theta_0 }. \]When to Use a Left-Sided Alternative
Suppose a manufacturer claims the average defect rate is no more than
\[ 2\%. \]If the study is designed to determine whether a revised process reduces the defect rate below \(2\%\), then the alternative can be
\[ H_A:p<0.02. \]Direction Comes from the Question
The direction of the alternative should be determined by the scientific, operational, or policy question.
It should not be chosen merely because the observed sample mean or sample proportion happens to fall on one side of the null value.
Choose the Alternative Before Looking at the Result
Selecting a one-sided alternative after seeing the data changes the probability structure of the test.
A valid directional test should be justified in advance by the research question.
One-Sided versus Two-Sided Evidence
A two-sided test allocates rejection probability to both tails of the null distribution.
A one-sided test allocates rejection probability to only one tail.
Therefore, one-sided and two-sided tests answer different questions and can produce different p-values.
Generic Parameter Notation
For a generic parameter \(\theta\), the three common alternatives are:
\[ H_A:\theta\neq\theta_0, \] \[ H_A:\theta>\theta_0, \] \[ H_A:\theta<\theta_0. \]The corresponding null hypothesis typically specifies
\[ H_0:\theta=\theta_0. \]Hypotheses for a Population Mean
For a population mean \(\mu\), common forms include:
\[ H_0:\mu=\mu_0, \]with one of
\[ H_A:\mu\neq\mu_0, \] \[ H_A:\mu>\mu_0, \]or
\[ H_A:\mu<\mu_0. \]Example: Mean Benchmark
A machine is calibrated to produce an average length of
\[ 20\text{ cm}. \]To determine whether the calibration has changed in either direction:
\[ \boxed{ H_0:\mu=20 } \] \[ \boxed{ H_A:\mu\neq20 }. \]Hypotheses for a Population Proportion
For a population proportion \(p\), common forms are
\[ H_0:p=p_0 \]versus one of
\[ H_A:p\neq p_0, \] \[ H_A:p>p_0, \]or
\[ H_A:pExample: Population Proportion
A company believes at least half of customers prefer a new design.
If the study is specifically intended to find evidence that the preference rate exceeds \(50\%\), a simple point-null formulation is
\[ H_0:p=0.50 \]versus
\[ H_A:p>0.50. \]Hypotheses for a Difference in Means
To compare two population means, the parameter of interest is often
\[ \mu_1-\mu_2. \]Equality of means corresponds to
\[ \mu_1-\mu_2=0. \]Thus a common two-sided test is
\[ H_0:\mu_1-\mu_2=0 \]versus
\[ H_A:\mu_1-\mu_2\neq0. \]Equivalent Two-Mean Notation
The statement
\[ \mu_1-\mu_2=0 \]is equivalent to
\[ \mu_1=\mu_2. \]Therefore, either form can express the same null hypothesis.
Difference notation is often useful because the test statistic is built directly from
\[ \bar{x}_1-\bar{x}_2. \]Directional Two-Mean Alternatives
If population 1 is expected to have a larger mean,
\[ H_A:\mu_1-\mu_2>0. \]Equivalently,
\[ H_A:\mu_1>\mu_2. \]If population 1 is expected to have a smaller mean,
\[ H_A:\mu_1-\mu_2<0. \]Hypotheses for a Difference in Proportions
To compare two population proportions, the parameter of interest is often
\[ p_1-p_2. \]A common two-sided test is
\[ H_0:p_1-p_2=0 \]versus
\[ H_A:p_1-p_2\neq0. \]Testing a Nonzero Difference
The null value does not always have to be zero.
Suppose a process change is considered meaningful only if the population mean increases by exactly \(5\) units relative to a benchmark model.
A null hypothesis could involve
\[ \mu_1-\mu_2=5. \]Hypothesis tests can be constructed around any specified null value that fits the question.
Scientific Claim versus Statistical Hypothesis
A scientific or business claim is often stated in ordinary language.
Statistical analysis translates that claim into parameter notation.
For example:
“The new system increases average throughput.”
can become
\[ H_A:\mu_{\text{new}}-\mu_{\text{old}}>0. \]Translate the Question Before Calculating
The hypotheses should be written before selecting formulas.
This prevents the arithmetic from driving the interpretation.
A useful workflow is:
- identify the population quantity;
- identify the benchmark or comparison;
- decide whether direction matters;
- write \(H_0\) and \(H_A\);
- then choose the test statistic.
Composite Hypotheses
A hypothesis is simple when it specifies a single parameter value.
A hypothesis is composite when it contains multiple possible parameter values.
For example,
\[ H_0:\mu=50 \]is simple, while
\[ H_A:\mu>50 \]is composite because it includes every value greater than \(50\).
Inequality Null Hypotheses
In more advanced testing problems, the null can itself be composite.
For example,
\[ H_0:\mu\le50 \]versus
\[ H_A:\mu>50. \]Many introductory tests calculate the reference distribution at the boundary value
\[ \mu=50. \]The exact treatment depends on the testing framework.
Boundary Values
In a one-sided composite null such as
\[ H_0:\mu\le\mu_0, \]the boundary
\[ \mu=\mu_0 \]often determines the critical reference case.
This is why introductory notation frequently writes the null using equality.
Claims of No Effect
In comparative studies, “no effect” often corresponds to a null difference of zero:
\[ H_0:\mu_1-\mu_2=0. \]For ratios, no effect may correspond to a value of \(1\) instead.
The correct null value depends on the parameter scale.
Null Values on Ratio Scales
Suppose the parameter is a risk ratio
\[ R. \]“No difference in risk” corresponds to
\[ \boxed{R=1}, \]not \(R=0\).
Similarly, an odds ratio of \(1\) represents no association on that scale.
Null Values Depend on the Parameter
Common null values include:
- \(0\) for differences;
- \(1\) for ratios;
- a benchmark mean such as \(100\);
- a benchmark proportion such as \(0.50\);
- a specified regression coefficient;
- a specified correlation value.
Hypotheses for a Correlation
Let
\[ \rho \]denote a population correlation coefficient.
A common test for no linear association is
\[ H_0:\rho=0 \]versus
\[ H_A:\rho\neq0. \]Hypotheses for a Regression Coefficient
Let
\[ \beta_1 \]be a population regression slope.
A common hypothesis test is
\[ H_0:\beta_1=0 \]versus
\[ H_A:\beta_1\neq0. \]The null says there is no linear slope effect in the specified model.
Hypotheses for a Variance
A population variance can also be tested.
For example,
\[ H_0:\sigma^2=25 \]versus
\[ H_A:\sigma^2\neq25. \]Under a normal population model, such tests are connected to the chi-square distribution.
The Alternative Determines the Tail
The hypothesis
\[ H_A:\theta>\theta_0 \]corresponds to a right-tailed test.
The hypothesis
\[ H_A:\theta<\theta_0 \]corresponds to a left-tailed test.
The hypothesis
\[ H_A:\theta\neq\theta_0 \]corresponds to a two-tailed test.
The Alternative Determines the Meaning of Extremeness
A test statistic of
\[ z=-2.5 \]can be strong evidence in a left-tailed test.
The same statistic is not evidence for a right-tailed alternative.
Statistical extremeness is always interpreted relative to the stated alternative.
Hypotheses Must Be Mutually Exclusive
The null and alternative should not both be true for the same parameter value.
For example,
\[ H_0:\mu=50 \]and
\[ H_A:\mu>45 \]overlap because \(\mu=50\) satisfies both statements.
A well-formed test avoids such overlap.
Hypotheses Should Cover the Relevant Possibilities
In many tests, the null and alternative together describe all relevant parameter values.
For example,
\[ H_0:\mu=50 \]versus
\[ H_A:\mu\neq50. \]Every possible value of \(\mu\) lies in one statement or the other.
Directional Claims Require Care
Suppose the practical claim is that a new treatment is better.
The statistical direction depends on how the outcome is measured.
If larger values are better,
\[ H_A:\mu_{\text{new}}>\mu_{\text{old}}. \]If smaller values are better, such as lower defect rates,
\[ H_A:p_{\text{new}}Words like Increase and Decrease Matter
Language in the research question often signals the direction.
Words such as
- increase;
- greater;
- higher;
- improve;
- reduce;
- lower;
- decrease;
should be translated carefully into parameter inequalities.
Words like Different Signal Two-Sided Tests
Words such as
- different;
- changed;
- not equal;
- deviates from;
often suggest a two-sided alternative.
The final choice should still reflect the actual study objective.
Example: Translating “Different”
Question:
“Has the average completion time changed from \(30\) minutes?”
Parameter:
\[ \mu. \]Hypotheses:
\[ \boxed{ H_0:\mu=30 } \] \[ \boxed{ H_A:\mu\neq30 }. \]Example: Translating “Greater”
Question:
“Is the average output greater than \(120\) units?”
Hypotheses:
\[ \boxed{ H_0:\mu=120 } \] \[ \boxed{ H_A:\mu>120 }. \]Example: Translating “Less”
Question:
“Has the defect proportion fallen below \(0.04\)?”
Hypotheses:
\[ \boxed{ H_0:p=0.04 } \] \[ \boxed{ H_A:p<0.04 }. \]Observed Statistics Do Not Determine the Hypotheses
Suppose the sample mean is
\[ \bar{x}=53. \]That does not automatically imply
\[ H_A:\mu>50. \]If the original question asked whether the mean differs from \(50\), the correct alternative remains
\[ H_A:\mu\neq50. \]The Null Is Not Necessarily What We Believe
The null hypothesis is a reference claim for the testing procedure.
It need not represent the researcher's personal belief.
Likewise, the alternative is not automatically true simply because it is the research claim.
The Null Is Not Automatically the Status Quo
Although the null often represents a status quo or no-effect benchmark, that is a convention rather than a universal law.
The null should be chosen to match the inferential structure of the problem.
Noninferiority and Equivalence Questions
Some applied questions do not fit the simple pattern
\[ H_0:\theta=\theta_0. \]In noninferiority and equivalence testing, the null can represent effects outside an acceptable margin.
These tests reverse some familiar roles and require careful formulation.
Equivalence Is Not the Same as Failure to Reject a Difference
Failing to reject
\[ H_0:\mu_1-\mu_2=0 \]does not prove the means are equivalent.
Equivalence requires a test specifically designed to show that differences are sufficiently small relative to a pre-specified equivalence margin.
Noninferiority
A noninferiority test asks whether a new method is not worse than a reference method by more than a chosen margin.
The margin must be justified before the analysis.
This is a different question from testing exact equality.
Hypotheses and Effect Sizes
A point null such as
\[ H_0:\mu_1-\mu_2=0 \]distinguishes zero from nonzero effects.
It does not by itself distinguish trivial effects from practically important effects.
Effect sizes and confidence intervals are therefore essential companions to hypothesis tests.
Hypotheses and Practical Thresholds
Suppose an increase of at least \(5\) units is required for business value.
Testing only
\[ H_0:\mu_1-\mu_2=0 \]answers whether the effect differs from zero.
It does not directly answer whether the effect exceeds the practical threshold of \(5\).
Hypotheses and Causal Claims
A hypothesis can be written about a difference between groups, but the statistical hypothesis alone does not establish causation.
Causal interpretation depends on the study design, such as random assignment or another defensible causal identification strategy.
Hypotheses and Generalization
Hypotheses may refer to a population parameter, but whether results generalize to that population depends on sampling and study design.
Correct notation cannot repair undercoverage, nonresponse, or selection bias.
Hypotheses and Model Assumptions
A null hypothesis is interpreted within a statistical model.
For example,
\[ H_0:\beta_1=0 \]in a linear regression model means something specific within that model.
If the model is seriously misspecified, the hypothesis-test interpretation may also be compromised.
Common Error: Putting the Sample Statistic in the Hypothesis
Incorrect:
\[ H_0:\bar{x}=50. \]Correct:
\[ \boxed{ H_0:\mu=50 }. \]Hypotheses concern population parameters, not observed sample summaries.
Common Error: Putting Equality in the Alternative
In standard point-null testing, writing
\[ H_A:\mu=50 \]is usually incorrect.
The equality belongs in the null hypothesis because that value defines the null reference model.
Common Error: Choosing the Tail from the Sample Result
If the sample mean happens to be above the null value, that does not justify changing a planned two-sided test into a right-sided test.
The alternative must reflect the original question.
Common Error: Using \(\neq\) When Only One Direction Matters
Suppose the actual question is whether a defect rate has decreased.
Writing
\[ H_A:p\neq p_0 \]tests for both increases and decreases.
If only a decrease is substantively relevant and the direction was specified in advance, the alternative should be
\[ H_A:pCommon Error: Using \(>\) and \(<\) in the Same Alternative
The statement
\[ H_A:\mu>50\text{ or }\mu<50 \]is mathematically equivalent to
\[ \boxed{ H_A:\mu\neq50 }. \]The inequality notation is cleaner and makes the two-sided structure explicit.
Common Error: Forgetting Which Group Is Subtracted
If the parameter is defined as
\[ \mu_1-\mu_2, \]then
\[ H_A:\mu_1-\mu_2>0 \]means population 1 has the larger mean.
Reversing the subtraction reverses the direction of the inequality.
Common Error: Confusing “No Evidence” with “Evidence of No Difference”
A standard test of
\[ H_0:\mu_1-\mu_2=0 \]that fails to reject does not establish practical equivalence.
Demonstrating equivalence requires a different hypothesis structure.
Common Error: Ignoring the Measurement Scale
“Higher is better” is not universal.
For defects, errors, response time, or risk, improvement may correspond to a lower numerical value.
The alternative must match the meaning of the variable.
A Reliable Procedure for Writing Hypotheses
- Identify the population or populations.
- Identify the parameter of interest.
- Translate the research question into a parameter comparison.
- Identify the benchmark or null value.
- Decide whether departures in one direction or both directions matter.
- Place the equality condition in the null for a standard point-null test.
- Write the alternative using \(<\), \(>\), or \(\neq\).
- Check that the hypotheses refer to population parameters, not sample statistics.
- Check that the null and alternative do not overlap.
- Confirm that the notation matches the substantive question.
Example: Full Mean-Hypothesis Workflow
A company claims that average delivery time is \(30\) minutes.
Management wants to know whether average delivery time has changed in either direction.
Parameter:
\[ \mu = \text{population mean delivery time}. \]Null hypothesis:
\[ \boxed{ H_0:\mu=30 }. \]Alternative hypothesis:
\[ \boxed{ H_A:\mu\neq30 }. \]Because either faster or slower delivery counts as a change, the test is two-sided.
Example: Full Proportion-Hypothesis Workflow
A quality team wants to determine whether the defect proportion has fallen below \(3\%\).
Parameter:
\[ p = \text{population defect proportion}. \]Null hypothesis:
\[ \boxed{ H_0:p=0.03 }. \]Alternative hypothesis:
\[ \boxed{ H_A:p<0.03 }. \]Because only a reduction is being tested, the alternative is left-sided.
Example: Full Two-Mean Workflow
A study compares average output under Method A and Method B and asks whether Method A produces greater average output.
Define
\[ \mu_A \]and
\[ \mu_B. \]Then
\[ \boxed{ H_0:\mu_A-\mu_B=0 } \]versus
\[ \boxed{ H_A:\mu_A-\mu_B>0 }. \]Practice 1: Mean, Two-Sided
A machine is supposed to produce an average weight of \(200\) grams. Investigators want to know whether the mean differs from \(200\).
Write \(H_0\) and \(H_A\).
Practice 2: Mean, Right-Sided
A new method is intended to increase average output above \(80\) units.
Write \(H_0\) and \(H_A\).
Practice 3: Proportion, Left-Sided
A process has historically produced a defect rate of \(5\%\). Researchers want to know whether the new process reduces that rate.
Write \(H_0\) and \(H_A\).
Practice 4: Difference in Means
Researchers want to know whether two population means are different.
Write the hypotheses using
\[ \mu_1-\mu_2. \]Practice 5: Difference in Proportions
Researchers want to know whether population 1 has a larger success proportion than population 2.
Write the hypotheses using
\[ p_1-p_2. \]Practice 6: Identify the Error
A student writes
\[ H_0:\bar{x}=40. \]What is wrong?
Practice 7: Choose the Tail
A study asks whether average waiting time has changed from \(15\) minutes, with increases and decreases both important.
Should the alternative be left-sided, right-sided, or two-sided?
Practice 8: Interpret Direction
Suppose
\[ H_A:\mu_1-\mu_2<0. \]Which population has the smaller mean under the alternative?
Practice 9: Ratio Null Value
If a risk ratio of \(1\) represents no difference in risk, what null value should be used when testing no difference?
Practice 10: Equivalence
A test fails to reject
\[ H_0:\mu_1-\mu_2=0. \]Does that prove the two population means are practically equivalent?
Practice Answers
Practice 1
\[ \boxed{ H_0:\mu=200 } \] \[ \boxed{ H_A:\mu\neq200 }. \]Practice 2
\[ \boxed{ H_0:\mu=80 } \] \[ \boxed{ H_A:\mu>80 }. \]Practice 3
\[ \boxed{ H_0:p=0.05 } \] \[ \boxed{ H_A:p<0.05 }. \]Practice 4
\[ \boxed{ H_0:\mu_1-\mu_2=0 } \] \[ \boxed{ H_A:\mu_1-\mu_2\neq0 }. \]Practice 5
\[ \boxed{ H_0:p_1-p_2=0 } \] \[ \boxed{ H_A:p_1-p_2>0 }. \]Practice 6
The null hypothesis should refer to a population parameter, not the sample mean.
A corrected form is
\[ \boxed{ H_0:\mu=40 }. \]Practice 7
The alternative should be two-sided:
\[ \boxed{ H_A:\mu\neq15 }. \]Practice 8
Because
\[ \mu_1-\mu_2<0, \]population 1 has the smaller mean:
\[ \boxed{\mu_1<\mu_2}. \]Practice 9
The null value is
\[ \boxed{1}. \]Practice 10
No.
Failure to reject an exact-difference null does not establish practical equivalence. Equivalence requires a procedure built around a pre-specified equivalence margin.
What to Remember
Null and alternative hypotheses define the logical structure of a hypothesis test.
- The null hypothesis provides the reference claim.
- The alternative hypothesis describes the departure of interest.
- Hypotheses should be written in terms of population parameters, not sample statistics.
- In standard point-null testing, equality belongs in the null hypothesis.
- \(\neq\) gives a two-sided alternative.
- \(>\) gives a right-sided alternative.
- \(<\) gives a left-sided alternative.
- The direction should be chosen from the research question before examining the result.
- Null values depend on the parameter scale: differences often use \(0\), while ratios often use \(1\).
- Failure to reject a null of no difference does not prove practical equivalence.
The most important habit is: translate the research question into a population parameter and decide the direction of the alternative before performing any test calculation.
Where We Go Next
Once the null and alternative hypotheses have been defined, the test still needs a rule for deciding how much evidence is enough to reject the null.
The next lesson, Significance Levels and P-Values, develops the significance threshold \(\alpha\), tail probabilities, p-values, rejection rules, and the correct interpretation of statistical evidence.