Statistics and Probability (COR 006)

Determining Sample Size for Estimating Population Mean and Proportion

Sample Size for Estimating Population Mean (μ)

• Confidence Level: We can be (1 - α) 100% confident that the error will not exceed a specified margin of error (e).
• Formula: Ensure sample size, n, is large enough by rounding up fractions to maintain the confidence level.
• Example: To be 90% confident that the error does not exceed 0.5 units with a standard deviation of 2, calculate the required sample size.

Sample Size for Estimating Population Proportion (p)

• Confidence Level: At least (1 - α)100% confident that the error does not exceed the margin of error (e).
• Example: To be 99% confident with an error margin within 0.02, determine the necessary sample size.

Hypothesis Testing

Definition

• Hypothesis: A statement or assertion about a phenomenon, which may be true or false.

Statistical Hypothesis

• An assumption or claim about a population's distribution.

Types of Hypotheses

• Null Hypothesis (Ho): States no relationship or difference exists; often includes terms like "not" or "no."
  • Example: No significant relationship between student attitudes and performance.
• Alternative Hypothesis (Ha): Opposes the null, suggesting a relationship or difference.
  • Example: Significant relationship between student attitudes and performance.

Types of Errors

• Type I Error (α): Incorrectly rejecting a true null hypothesis.
• Type II Error (β): Failing to reject a false null hypothesis.

One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: Directional, used when Ha involves "greater than" or "less than."
• Two-Tailed Test: Non-directional, used when Ha indicates "not equal to."

Rejection Regions

• Defined areas in the distribution where the null hypothesis is rejected.
• Critical Values: Boundary points that determine rejection of Ho.

Steps in Hypothesis Testing

1. Formulate Hypotheses: Define the null and alternative hypotheses.
2. Identify Significance Level (α) and Test Statistic:
   • Common levels: 0.01, 0.05, 0.10.
3. Determine Critical Value: Use tables to find critical values for one- or two-tailed tests.
4. Decision Rule: Accept or reject Ho based on comparison between test statistic and critical value.
5. Conclusion: Present findings and support with evidence.

Identifying Appropriate Test Statistics

• Z-Test: Used with known population standard deviation and large samples (n ≥ 30).
• T-Test: Used with unknown population standard deviation and small samples (n < 30).
• F-Test: Compares variances across multiple populations, applicable in ANOVA.

Example

• Example of a Z-Test scenario with known population standard deviation.

Central Limit Theorem

• Applied when sample size is large (n ≥ 30), regardless of known population variance.
• Supports using the normal distribution for hypothesis testing.