Comprehensive Guide to Hypothesis Testing

Overview of Hypothesis Testing

• Definition of Statistical Hypothesis: A statistical hypothesis is a claim or statement regarding the value or values of a specific population parameter.

• The Statistical Process: To verify claims statistically, sample data must be gathered and analyzed. The results from the sample lead to one of two conclusions about the hypothesized value of the population parameter:

  • The value is accepted as probably true.

  • The value is rejected as probably false.

• Hypothesis Testing Definition: This is the formal statistical process used to test the validity of a claim regarding the true value of any population parameter.

Types of Hypothesis

• 1. Null Hypothesis ($H_{0}$):

  • This is the specific hypothesis that is being tested.

  • It represents a claim or statement about a population parameter that is assumed to be true until evidence suggests it is false.

  • It is often called the "hypothesis of no difference."

  • Notation: Includes symbols of equality or semi-equality, such as $H_{0}: (=, \text{ or } \text{ ≥, } \text{ or } \text{ ≤})$.

• 2. Alternative Hypothesis ($H_{1}$):

  • This is the hypothesis believed to be true if the null hypothesis ($H_{0}$) is rejected.

  • It is known as the "hypothesis of difference."

  • Notation: Includes symbols of inequality, such as $H_{1}: (\neq, \text{ or } <, \text{ or } >)$.

Directions and Tails in Hypothesis Testing

• Rejection Region Logic: The rejection region of a test is determined entirely by the alternative hypothesis ($H_{1}$).

• Two-Tailed Test: Defined by the symbol $\neq$ in the alternative hypothesis. This test has two rejection regions (one in each tail of the distribution).

• One-Tailed Test: Defined by the symbols $>$ or $<$ in the alternative hypothesis. These tests have only one rejection region.

  • One-Sided Upper-Tailed Test: Used when the claim states the parameter is more than a specified value.

    • $H_{0}: \text{population parameter} \text{ ≤ } \text{specified value}$

    • $H_{1}: \text{population parameter} > \text{specified value}$

    • The rejection region is in the far right tail.

  • One-Sided Lower-Tailed Test: Used when the claim states the parameter is less than a specified value.

    • $H_{0}: \text{population parameter} \text{ ≥ } \text{specified value}$

    • $H_{1}: \text{population parameter} < \text{specified value}$

    • The rejection region is in the far left tail.

Risks and Errors in Hypothesis Testing

Because hypothesis tests are based on samples (which may be biased or non-representative), there is always a probability of making an incorrect decision.

• Type I Error ($α$): The probability of rejecting the null hypothesis when it is actually true.

  • $α = P(\text{rejecting } H_{0} | H_{0} \text{ is true})$

• Type II Error ($β$): The probability of accepting (not rejecting) the null hypothesis when it is actually false.

  • $β = P(\text{not rejecting } H_{0} | H_{0} \text{ is false})$

Decision Matrix Summary:

• Relationship between $α$ and $β$: These probabilities work inversely; as the probability of a Type I error ($α$) increases, the probability of a Type II error ($β$) decreases.

Hypothesis Test for a Single Population Mean ($μ$)

Hypotheses Formulation

• Left-tailed: $H_{0}: μ \text{ ≥ } μ_{0}$; $H_{1}: μ < μ_{0}$

• Right-tailed: $H_{0}: μ \text{ ≤ } μ_{0}$; $H_{1}: μ > μ_{0}$

• Two-tailed: $H_{0}: μ = μ_{0}$; $H_{1}: μ \neq μ_{0}$

Test Statistics

• Z-Test Statistic: Used when the population standard deviation ($σ$) is known or the sample size is large ($n \text{ ≥ } 30$).

  • $Z_{calc} = \frac{\bar{x} - μ_{0}}{\frac{σ}{\text{√}n}}$

  • Critical Value: $Z_{crit} = Z_{table} = Z_{α}$

• T-Test Statistic: Used when the population standard deviation ($σ$) is unknown and the sample size is small ($n < 30$).

  • $t_{calc} = \frac{\bar{x} - μ_{0}}{\frac{s}{\text{√}n}}$

  • Denominator Note: $\frac{s}{\text{√}n}$ is an estimate for the standard error term $\frac{σ}{\text{√}n}$.

  • Critical Value: $t_{crit} = t_{table} = t_{α, df}$ where $df = n - 1$.

Distribution Consistency Table

Hypothesis Test for Single Population Proportion ($π$)

Hypotheses Formulation

• Left-tailed: $H_{0}: π \text{ ≥ } π_{0}$; $H_{1}: π < π_{0}$

• Right-tailed: $H_{0}: π \text{ ≤ } π_{0}$; $H_{1}: π > π_{0}$

• Two-tailed: $H_{0}: π = π_{0}$; $H_{1}: π \neq π_{0}$

Calculation

• Sample Proportion ($p$): $p = \frac{x}{n}$, where $x$ is the number of successes in sample size $n$.

• Z-Test Statistic:

  • $Z_{calc} = \frac{p - π}{\text{√}\frac{π(1 - π)}{n}}$

• Critical Value: $Z_{crit} = Z_{table} = Z_{α}$

Hypothesis Test for the Difference Between Two Population Means ($μ_{1} - μ_{2}$)

Hypotheses Formulation

• Left-tailed: $H_{0}: μ_{1} \text{ ≥ } μ_{2}$; $H_{1}: μ_{1} < μ_{2}$

• Right-tailed: $H_{0}: μ_{1} \text{ ≤ } μ_{2}$; $H_{1}: μ_{1} > μ_{2}$

• Two-tailed: $H_{0}: μ_{1} = μ_{2}$; $H_{1}: μ_{1} \neq μ_{2}$

Calculation Cases

• Case 1: $σ_{1}, σ_{2}$ known; $n_{1}, n_{2}$ large.

  • $z_{calc} = \frac{\bar{x}_{1} - \bar{x}_{2}}{\text{√}(\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}})}$

• Case 2: $σ_{1}, σ_{2}$ known; $n_{1}, n_{2}$ small.

  • Same Z-statistic formula as Case 1.

• Case 3: $σ_{1}, σ_{2}$ unknown; $n_{1}, n_{2}$ large.

  • $z_{calc} = \frac{\bar{x}_{1} - \bar{x}_{2}}{\text{√}(\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}})}$

• Case 4: $σ_{1}, σ_{2}$ unknown; $n_{1}, n_{2}$ small.

  • Assumption: Population variances are equal ($σ_{1}^{2} = σ_{2}^{2} = σ^{2}$).

  • Pooled Variance ($S_{p}^{2}$):

    • $S_{p}^{2} = \frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}$

  • T-Test Statistic:

    • $t_{calc} = \frac{\bar{x}_{1} - \bar{x}_{2}}{\text{√}(S_{p}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}}))}$

  • Critical Value: $t_{α, n_{1} + n_{2} - 2}$

Examples and Numerical Case Studies

Single Mean Examples

• Example 1 (SARS Tax Returns): SARS claims eFiling takes typical taxpayers less than 45 minutes on average ($μ < 45$). Sample of $n = 12$. Times: 42, 56, 29, 35, 47, 37, 39, 29, 45, 35, 51, 53. Sample mean $\bar{x} = 41.5$, sample SD $s = 9.04$. Test at $α = 0.05$.

• Example 2 (Cement Bags): Hardware store receives complaints that "1 kg" bags weight less than 1 kg ($μ < 1$). Sample $n = 40$, mean $\bar{x} = 0.987$, known population SD $σ = 0.025$. Test at $α = 0.05$.

• Example 3 (Coffee Containers): Manager suspects over-filling of "500 g" containers ($μ > 500$). Sample $n = 30$, mean $\bar{x} = 501.8$, known population variance $σ^{2} = 60$ ($σ = \text{√}60$). Test at $α = 0.05$.

• Example 4 (Frozen Shrimp Cans): Factory manager checks if cans conform to label weight of 600 g ($μ = 600$). Tests for both over and under filling ($μ \neq 600$). Sample $n = 50$, mean $\bar{x} = 595$, known population SD $σ = 20$. Test at $α = 0.05$ and $α = 0.10$.

Single Proportion Examples

• Example 1 (Construction Schedule): Company suspects 20% of jobs are behind schedule ($π = 0.2$). Data: $n = 80$, $x = 22$ late jobs. Calculation: $p = \frac{22}{80} = 0.275$. Test at $α = 0.05$.

• Example 2 (Product Marketing): Campaign for new product. Data: $x = 176$ out of $n = 200$ users would use it ($p = \frac{176}{200} = 0.88$). Test evidence for more than 85% ($π > 0.85$) at $α = 0.01$.

Two Means Examples

• Example 1 (Hospital Response): Comparison of response times for senior citizens vs other patients. Complaint: Seniors take longer ($μ_{senior} > μ_{other}$). Test at $α = 0.01$.

• Example 2 (Metal Hardening): Comparing saltwater quenching ($μ_{1}$) vs oil quenching ($μ_{2}$). Assume hardness is normally distributed and variances are equal. Test $H_{0}: μ_{1} = μ_{2}$ at $α = 0.05$.