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StatisticsConcept reference

Null hypotheses

A null and alternative are statements about a population parameter; complementary hypotheses divide the allowed parameter values into the reference claim and the investigated alternative.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Hypotheses state claims about a population parameter
  3. For a population mean μ, “different from 6” gives H_a:μne6 and H_0:μ=6
  4. A closer look
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Hypotheses state claims about a population parameter#

Hypotheses state claims about a population parameter. Here HaH_a is the direction being investigated; H0H_0 contains the remaining allowed values. These complementary hypotheses never overlap and cover the parameter model.

Suppose θ\theta can be 1,2,3,41,2,3,4. To investigate θ>2\theta>2, use H0:θ≤2H_0:\theta\le2 and Ha:θ>2H_a:\theta>2. Equality belongs to the null.

In this illustrative model θ can equal only 1,2,3,4. H0 states θ≤2 and contains 1,2. Ha states θ>2 and contains 3,4. Every permitted parameter value belongs to exactly one hypothesis, and 2 belongs to the null.In this illustrative model θ can equal only 1,2,3,4. H0 states θ≤2 and contains 1,2. Ha states θ>2 and contains 3,4. Every permitted parameter value belongs to exactly one hypothesis, and 2 belongs to the null.
Figure 1In this illustrative model θ can equal only 1,2,3,4. H0 states θ≤2 and contains 1,2. Ha states θ>2 and contains 3,4. Every permitted parameter value belongs to exactly one hypothesis, and 2 belongs to the null.
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Check your reasoning

μ: population mean. Investigate μ below 5. Complementary H0 / Ha?

  1. Aμ≥5 / μ<5
  2. Bμ≤5 / μ>5
  3. Cμ=5 / μ≠5
Show answer and explanation
μ≥5 / μ<5

Ha: μ<5; H0 is its complement.

02For a population mean μ, “different from 6” gives H_a:μne6 and H_0:μ=6#

For a population mean μ\mu, “different from 6” gives Ha:μ≠6H_a:\mu\ne6 and H0:μ=6H_0:\mu=6. The sample mean xˉ\bar x is evidence about μ\mu, not the parameter named in these hypotheses.

Check your reasoning

p: population proportion; p̂: sample value 0.4. Investigate p above 0.3. Complementary H0 / Ha?

  1. Ap̂≤0.3 / p̂>0.3
  2. Bp≤0.3 / p>0.3
  3. Cp=0.3 / p≠0.3
Show answer and explanation
p≤0.3 / p>0.3

Use population p.

03A closer look#

A population proportion pp describes the target population; an observed proportion p^\hat p summarizes a sample. Use the question’s reference value and predeclared direction. An observed value above the reference does not change a predeclared “below” investigation.

Check your reasoning

μ: population mean; x̄: sample value 12. Investigate μ different from 10. “Use x̄.” Complementary H0 / Ha?

  1. Ax̄=10 / x̄≠10
  2. Bμ≥10 / μ<10
  3. Cμ=10 / μ≠10
Show answer and explanation
μ=10 / μ≠10

Use population μ.

Key takeaway

Name the population parameter, put the predeclared investigation in the alternative, and assign the remaining values to the null.

  • Specify a complementary null and alternative.

Sources & further reading

  1. [1]
    Pishro-Nik, 8.4.2 General Setting and Definitions ↗Pishro-Nik, Introduction to Probability · Book
  2. [2]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Null hypotheses.” Math behind ML. /learn/s-null