Hypothesis Testing - Study Notes

HYPOTHESIS TESTING

  • Hypothesis testing is a statistical method used in making decisions based on experimental data. It is basically testing an assumption about a population.

  • The method helps decide whether observed data provide enough evidence to reject a claim about the population parameter or the distribution of a random variable.

HYPOTHESIS

  • A HYPOTHESIS is a proposed explanation, assertion, or assumption about a population parameter or about the distribution of a random variable.

  • It identifies what we want to learn about the population and what value we believe the parameter may take (the HYPOTHESIZED VALUE).

KEY CONCEPTS IN HYPOTHESIS TESTING

  • POPULATION parameter: a numeric characteristic of the population (e.g., mean μ, proportion p).

  • HYPOTHESIS: a claim about the population parameter or distribution; tested using sample data.

  • HYPOTHESIZED VALUE: the specific value of the parameter that is proposed in the hypothesis (often denoted μ0 or p0).

  • VALUES BEING COMPARED: the hypothesized value vs. the actual population parameter inferred from data.

  • CLAIM ABOUT THE PARAMETER: the assertion expressed in the hypothesis about what the parameter equals or how it compares to a value.

EXAMPLES OF HYPOTHESES

  • Example 1: The average monthly income of Filipino families in the low-income bracket is Php 9,000.

    • POPULATION: the group of low-income Filipino families

    • PARAMETER: population mean income μ

    • HYPOTHESIZED VALUE: μ0 = Php 9,000

  • Example 2: The average height of adult Filipinos aged 20+ is 163 cm.

    • POPULATION: adult Filipinos 20 years and older

    • PARAMETER: population mean height μ

    • HYPOTHESIZED VALUE: μ0 = 163 cm

NULL AND ALTERNATIVE HYPOTHESES

  • NULL HYPOTHESIS (Ho): an initial claim based on prior analyses which the researcher tries to disprove, reject, or nullify. It typically states no significant difference or no effect.

  • DENOTATION: Ho.

  • ALTERNATIVE HYPOTHESIS (Ha): the claim opposite to Ho; indicates observations are the result of a real effect.

  • DENOTATION: Ha.

  • EXAMPLE: The average number of vehicles passing through NLEX daily is less than 21,000.

    • Ho: μ ≥ 21{,}000 (the average is at least 21,000 vehicles/day – status quo)

    • Ha: μ < 21{,}000 (the average is less than 21,000)

    • Note: In many texts Ho is framed as an equality or as ≥/≤ depending on the test direction; Ha represents the claim to be proved.

  • EXAMPLE: The school record claims that the mean Math score of incoming Grade 11 students is 81. The teacher tests whether there is a significant difference between the batch mean and the class mean.

    • Ho: μ = μ0 (mean score equals the claimed mean)

    • Ha: μ ≠ μ0 (mean score is different from the claimed mean)

  • SYMBOLS TO REMEMBER:

    • μ = population mean

    • μ0 = hypothesized mean value

    • x̄ = sample mean

    • p̂ = sample proportion

    • p0 = hypothesized population proportion

    • ρ or σ^2 = population variance (depending on context)

FORMS OF HYPOTHESES

  • For a two-sided (two-tailed) test: Ha: μ ≠ μ0

    • Ho: μ = μ0

  • For a one-sided (one-tailed) test:

    • If Ha: μ < μ0 → Ho: μ ≥ μ0

    • If Ha: μ > μ0 → Ho: μ ≤ μ0

  • More generally, hypotheses can be framed as Ho: parameter equals or is bounded, and Ha as the corresponding inequality or ≠ depending on the claim.

LEVEL OF SIGNIFICANCE (ALPHA)

  • α denotes the degree of significance, i.e., the probability of making a Type I error when Ho is true.

  • Common values: α = 0.01 (1%), α = 0.05 (5%), α = 0.10 (10%).

  • It is not possible to be 100% certain when making a decision about Ho.

  • If Ha uses ≠, then α is split between the two tails: α/2 in each tail.

  • PRACTICAL GUIDANCE: In public health studies, α is often 0.01; in social sciences, α is often 0.05 or 0.10.

  • EXAMPLE: Maria uses a 5% level of significance to test whether there is no significant change in the average number of enrollees across last two years. This means that the probability of rejecting Ho when it is true is 5% (α = 0.05).

  • FURTHER NOTE: When Ha is two-tailed (μ ≠ μ0), the critical region is split across both tails, so the total α is divided as α/2 in each tail.

TWO-TAILED VS ONE-TAILED TESTS

  • TWO-TAILED TEST (non-directional): Ha: μ ≠ μ0

  • ONE-TAILED TEST (directional):

    • Ha: μ < μ0 (left-tailed)

    • Ha: μ > μ0 (right-tailed)

  • DECISION CRITERIA depend on the test type and the chosen α.

  • EXAMPLE: The school registrar believes the average number of enrollees this school year is not the same as the previous year. Let μ0 be the previous year's average.

    • Two-sided interpretation: Ho: μ = μ0; Ha: μ ≠ μ0

    • One-sided interpretation: if the registrar suspects a decrease, Ha: μ < μ0 ( Ho: μ ≥ μ0 ), etc.

REJECTION REGION AND CRITICAL VALUES

  • Rejection region (critical region): set of all values of the test statistic that lead to rejecting Ho.

  • Non-rejection (acceptance) region: set of values that fail to reject Ho.

  • Critical value: boundary value on the test distribution that separates rejection and non-rejection regions.

  • One-tailed test: rejection region is on one side of the distribution.

    • Left-tailed: statistic ≤ -tα,df (or zα,df for z-tests)

    • Right-tailed: statistic ≥ tα,df (or zα,df)

  • Two-tailed test: rejection regions are in both tails beyond ±tα/2,df (or ±zα/2).

CRITICAL VALUES (Z-TEST TABLE)

  • For z-tests (normal approximation), the critical values depend on α and whether the test is one-tailed or two-tailed:

  • One-tailed test:

    • α = 0.05 → z-critical ≈ ±1.645 (right-tailed: +1.645; left-tailed: -1.645)

    • α = 0.01 → z-critical ≈ ±2.33

    • α = 0.001 → z-critical ≈ ±3.30

  • Two-tailed test (split α/2 in each tail):

    • α = 0.05 → z-critical ≈ ±1.96

    • α = 0.01 → z-critical ≈ ±2.58

    • α = 0.001 → z-critical ≈ ±3.30

  • Note: When using t-tests, use t-critical values tα,df that depend on the degrees of freedom df = n−1 (or df appropriate for the test).

PROPORTION AND MEAN TEST EXAMPLES

  • PROPORTION TEST EXAMPLE (TRYIT-style)

    • Problem setup: Compare a sample proportion p̂ to a hypothesized proportion p0 at significance α.

    • Test statistic (for large n):
      z=racp^−p<em>0p</em>0(1−p0)/nz = rac{ \hat{p} - p<em>0 }{ \sqrt{ p</em>0(1 - p_0)/n } }

    • EXAMPLE SETUP (from slides): 2015 data: p0 = 0.34; n = 500; observed p̂ = 0.18.

    • Compute SE: p<em>0(1−p</em>0)/n=sqrt0.34⋅0.66/500approx0.0212\sqrt{ p<em>0(1 - p</em>0)/n } = \\sqrt{ 0.34 \cdot 0.66 / 500 } \\approx 0.0212

    • Compute z-score: z=(0.18−0.34)/0.0212≈−7.56z = (0.18 - 0.34) / 0.0212 \approx -7.56

    • Critical value at α = 0.05 (two-tailed for difference): z_{0.025} ≈ 1.96; since |-7.56| > 1.96, reject Ho strongly: there is evidence that the current percentage differs from 34%. (
      Note: The slide labels and interpretation may vary by context; this is the standard large-sample proportion test approach.)

  • MEAN TEST EXAMPLE (t-test)

    • Given: $\mu$ (hypothesized mean), sample mean $\bar{x}$, sample standard deviation $s$, sample size $n$.

    • Test statistic (one-sample t):
      t=xˉ−μ0s/nt = \frac{ \bar{x} - \mu_0 }{ s / \sqrt{n} }

    • Example from slides: $\mu = 142$ (global mean), $\bar{x} = 152$, $s = 19.855$, $n = 10$.

    • Compute: t=152−14219.855/10=106.277≈1.593t = \frac{152 - 142}{19.855 / \sqrt{10}} = \frac{10}{6.277} \approx 1.593

    • Degrees of freedom: $df = n - 1 = 9$.

    • Critical value for a one-tailed test at α = 0.05: $t_{0.05,9} \approx 1.833$.

    • Decision: computed $t$ (1.593) < 1.833, so do not reject Ho at α = 0.05 for a one-tailed test; conclude there is not enough evidence that the sample mean differs in the specified direction.

DECISIONS AND ERRORS IN HYPOTHESIS TESTING

  • TYPE I ERROR (false positive): Rejecting Ho when Ho is true.

    • Probability of a Type I error is α.

    • The normal curve with the rejection region is called the alpha region.

  • TYPE II ERROR (false negative): Failing to reject Ho when Ho is false.

    • Probability of a Type II error is β.

    • The normal curve with the acceptance region is called the beta region.

  • RELATIONSHIP: Increasing α generally lowers β (higher chance of finding a statistical effect when one exists) but increases the risk of a false positive.

  • SUMMARY OF DECISIONS:

    • If Ho is true and we reject Ho → Type I error.

    • If Ho is false and we fail to reject Ho → Type II error.

    • If Ho is true and we fail to reject Ho → correct decision.

    • If Ho is false and we reject Ho → correct decision.

  • CHART-STYLE ILLUSTRATIONS (as described in slides):

    • Region where Ho is true corresponds to the Type I error region.

    • Region where Ho is false corresponds to the Type II error region.

PRACTICAL ILLUSTRATIONS OF ERRORS (EXAMPLES)

  • Example 1 (Type I error): A person claims to be 30 years old; the true age is 32. If a test concludes the mean age is not 30 when it is actually 32, that would be a rejection of Ho when Ho is false in this real-world framing. (Note: Standard definition says Type I is rejecting Ho when Ho is true; real-world examples may be framed differently on the slide.)

  • Example 2 (Type I error): A claim of not being bald is rejected when the person’s hairline is actually receding. (Again, standard interpretation depends on how Ho is framed in the test.)

  • Example 3 (Type II error): Believing that a rare event does not occur (e.g., a dangerous species absence) when it actually does occur; failing to detect the effect.

  • Example 4 (Type I vs Type II in retirement mean): If the mean years of service to retirement is claimed to be 30 years, testing this claim and concluding it is not 30 when, in fact, the true mean is 30 can be described as a Type I error under some phrasing. The slides also present a Type II error scenario with mis-stated mean values.

  • NOTE ON EXAMPLES: The slides include practical sentences illustrating how to phrase Type I and Type II errors. In standard statistical terminology, Type I error occurs when Ho is true and we reject it; Type II error occurs when Ho is false and we fail to reject it. Use the standard definitions when solving problems, and align the decision rules (rejection vs non-rejection) with the chosen α and the test statistic.

SUMMARY OF KEY TAKEAWAYS

  • Hypothesis testing involves Ho (null) and Ha (alternative) about a population parameter.

  • Ho is typically an equality or an inequality that reflects the default status quo; Ha describes what you aim to show.

  • Tests can be one-tailed or two-tailed depending on the research question and Ha.

  • The level of significance α is the probability of a Type I error; common values are 0.05, 0.01, 0.10.

  • The rejection region is defined by critical values (z or t) determined by α and df, and whether the test is left-tailed, right-tailed, or two-tailed.

  • For mean testing with known variance, use a z-test; otherwise, use a t-test with df = n − 1.

  • For proportion testing, large-sample z-tests are used with the formula for z given above.

  • Type I error (α) vs Type II error (β) are inversely related in general; increasing α lowers β and vice versa.

  • Always clearly state the hypotheses, the test statistic, the rejection region, and the conclusion in terms of Ho or Ha.

ADDITIONAL NOTES (RECOMMENDED)

  • When Ha is two-sided (Ha: μ ≠ μ0), split α into α/2 in each tail of the distribution to determine the rejection regions.

  • When Ha is one-sided, place the entire α in the relevant tail.

  • In interpreting results, also consider practical significance and study design limitations, not solely statistical significance.

  • If performing a proportion test with small expected counts, consider a different test (e.g., exact binomial) instead of the normal approximation.