Study Notes on Chapter 5: Inferences on Population Means

Chapter 5: Inferences On One Or Two Population Means

Central Limit Theorem: For a large sample size (n ≥ 25-30), the sample mean (X) is approximately normally distributed with mean (μ).
Confidence Intervals:
About 95.45% of the time, the sample mean will lie within the interval: μ ± 2σ/√n
About 99.7% of the time, it will lie within μ ± 3σ/√n
Confidence Level:
Denoted by (1 - α), where α is the significance level (e.g., 0.05 for a 95% confidence level).
As (1 - α) approaches 1, confidence in the interval containing the mean increases.
The critical value zα/2 corresponds to the (1 - α/2) × 100th percentile of the standard normal distribution.

Population mean: μ = 266 days, standard deviation: σ = 16 days.
Simulation conducted using R for 10,000 confidence intervals (n = 40) at α = 0.05.

[ E = z_{α/2} \frac{σ}{\sqrt{n}} ]

Minimum required sample size for given margin of error E:
[ n = \left( \frac{z_{α/2} · σ}{E} \right)^2 ]

Margin of error E = 9 days, standard deviation σ = 16 days, for a 95% CI.
Required sample size calculation to remain within the margin of error covers how sample size relates to confidence intervals.

As the margin of error increases, the width of the confidence interval increases.
As the confidence level decreases, the width of the interval decreases.
As sample size increases, the width of the interval decreases.

Assume H₀ is true, take a sample.
Compute a test statistic.
Determine if the test statistic is in the rejection region (RR) to reject H₀ or not.

Example 56: Your professor's claim of an IQ of at least 140 tested with sample statistics showing if various observed IQs lead to rejection or acceptance of H₀.

When population standard deviation (σ) is known, conduct right-tailed, left-tailed, or two-tailed tests based on sample mean (X̄).
Test Statistics Calculations:
Right-tailed test:
- H₀: μ ≤ μ₀, H₁: μ > μ₀
Left-tailed test builds on similar methodology.
Compute p-value: probability of obtaining the observed test statistic given that H₀ is true.
Examples 57 & 58 detail specific case testing for soybean yields and cholesterol levels in immigrants, illustrating test statistic computation and resultant p-values leading to conclusion statements.

The goal is to achieve small values for both α and β (type II error). Minimum required sample size for Type II error manageable based on hypothesis testing model adjusted to save time and resources.
The formula adjusts based on the desired power and β considerations,
[ n = \left( \frac{Z{a} + Z{β}}{μ₁ - μ₀} \right)^2 ]

When σ is unknown, use the t-distribution for hypothesis testing and confidence intervals:
The behavior of the sample mean approaches normality as n increases thanks to the Central Limit Theorem.
Key Formula:
[ T = \frac{X̄ - μ₀}{S/\sqrt{n}} ] where S is the sample standard deviation.
Example 63: Salamander Lengths: Illustrates the use of t-distribution and subsequently computes the 95% confidence interval finding actual mean of salamander length using sample data.

The t-distribution is used when sampling from normal distributions with small samples or large samples where σ is unknown.
As degrees of freedom increase, the t-distribution approaches a normal distribution.
Conclusions drawn from hypothesis testing depend heavily on p-values, which signify the strength of the evidence against H₀.