Chapter 5 Notes

QBIO 305 Statistics for the Life Sciences - Study Notes

Meta-Study

• Definition: A meta-study (or meta-analysis) combines results from multiple independent studies.
• Components:
  • Involves many repetitions or replications of the same study.
  • If a study consists of drawing a random sample of size n from a population:
  • The meta-study involves drawing repeated random samples of size n from the same population.
• Goals:
  • Increase sample size.
  • Reduce random noise.
  • Obtain a more reliable overall conclusion.

Sampling Variability

• Definition: When repeatedly taking random samples from the same population:
  • Each sample will yield slightly different results.
  • Sample means and sample proportions vary from sample to sample.
• Nature of Sampling Variability:
  • Exists even when there is no measurement error.
  • Present when a study is conducted correctly.

Sampling Distribution

• Definition: Describes how a statistic behaves.
• Procedure:
  1. Take a random sample.
  2. Compute a statistic (e.g., mean, proportion).
  3. Repeat many times.
• Examples:
  • Distribution of sample means, denoted as $\bar{X}$.
  • Distribution of sample proportions, denoted as $ilde{p}$.

Key Theorems: LLN and CLT

• Focus on two key theorems in statistics:
  • Law of Large Numbers (LLN): As the sample size n increases, the sample mean $\bar{X}_n$ converges to the population mean $ext{µ}$.
  • Central Limit Theorem (CLT): For large sample sizes, the sum (or average) of independent random variables will tend to follow a normal distribution regardless of the original distribution of the variables.

Simple Random Sample

• Definition: Each member of the population has an equal chance of being included in the sample.
• Characteristics:
  • Sample members are chosen independently of one another.

i.i.d Distribution

• Definition: In a simple random sample, elements are said to be independent and identically distributed (i.i.d), meaning:
  • They come from the same probability distribution.

Law of Large Numbers (LLN)

• Notation: Let $X<em>1, X</em>2,…, X_n$ be independent and identically distributed random variables, each with mean $ext{µ}$.
• Statement: As the sample size $n$ approaches infinity:
  • $\bar{X}<em>n = \frac{\sum</em>{i=1}^{n} X_i}{n} \rightarrow \text{µ}$.
• Interpretation: The sample mean converges to the population mean.

Intuitive Understanding

• Conceptual Understanding: As sample size becomes arbitrarily large, the sample mean becomes arbitrarily close to the population mean.

Coin Flipping Example

• Experiment: Flipping a fair quarter 100 times to determine the expected number of heads.
• Distribution of Results:
  • Let: $X_i = 1$ with probability $\frac{1}{2}$ (Heads).
  • Let: $X_i = 0$ with probability $\frac{1}{2}$ (Tails).
• Calculated Mean: $ext{µ} = E[X_i] = 1 \cdot \frac{1}{2} + 0 \cdot \frac{1}{2} = 0.5$.
• Conclusion: The Law of Large Numbers indicates that the sample mean will converge to $0.5$.

Bernoulli Trials Example

• Consideration: Flipping a coin, a random variable $X_i$ results in 1 (success) or 0 (failure).
• Mean: $E[X_i] = p$.
• LLN: Sample means will converge to the probability of success p.

Expected Value Interpretation

• Frequentist Interpretation: The probability of an event occurring with probability $p$ is interpreted as the frequency of that event happening in many independent trials.

Rolling a Die Example

• Random Variable: $X_i$ representing die results.
• Mean Calculation:
  • $E[X_i] = \frac{1+2+3+4+5+6}{6} = 3.5$.
• LLN Application: Sample means will converge to 3.5 as sample size increases.

Central Limit Theorem (CLT)

• Definition: Let $X<em>1, X</em>2,…, X_n$ be independent and identically distributed random variables with mean $µ$ and standard deviation $σ$.
• Asymptotic Behavior: For large n:
  • $\frac{\sum<em>{i=1}^{n}(X</em>i - \text{µ})}{\sigma \sqrt{n}} \rightarrow N(0, 1)$ (standard normal distribution).

Convergence of Random Variables

• Interpretation: The distribution functions of the two random variables converge, establishing a link between sample distributions and normal distributions.

Graphical Representation of Sampling Distribution

• Illustration: Based on means from sampling sizes (n=4, n=7, n=10), with a consistent trend showing convergence to normality as n increases.

Galton Board Example

• Summary: The final position of a ball in a Galton board represents the sum of independent random movements (approximation to normal distribution due to the Central Limit Theorem).

Normal Distribution Properties

• Property: For a normal random variable $Z$ with mean $µ$ and standard deviation $σ$, the transformed random variable $aZ + b$ is also normal with:
  • Mean: $aµ + b$.
  • Standard Deviation: $|a|σ$.

Behavior of Summed Random Variables

• For large sample sizes: $\frac{\sum<em>{i=1}^{n}X</em>i}{n} \to N(\text{nµ}, \text{σ})$ for sum of independent random variables.
• Conclusion: The sum of many independent variables leads to a normal distribution, explaining phenomena such as height variation in populations.

Population Height Distribution

• Graphical Data: U.S. Height Distribution for men and women illustrating normal distribution characteristics among large populations.

Blood Pressure Example

• Frequency Distribution: Diastolic blood pressure data shown in a histogram format indicating overall population normality and behavior.

Total Cholesterol Distribution Example

• Utilizes statistical analysis for age ≤ 20 demonstrating stable distribution with various statistics reported.

Application of the Central Limit Theorem in Real-World Scenarios

• Baseball Batting Average: Demonstrates averaging outcomes of numerous independent, binomially distributed at-bats, yielding normal distribution characteristics for sample means.

Implications of Sample Size on Distribution

• Clarification: Sample size influences the average closer to normality but does not alter the underlying population distribution itself.
• Common Misconception: Increasing sample size does not convert non-normal distributions to normal. Instead, only the distribution of sample means becomes normal due to CLT principles.

Roulette Game Example

• Game Structure: Standard American roulette outlined, including expected gains correlating to the casino's advantages and expected negative returns for players over time.
• Statistical Expectations: In the long run, players will experience negative profits due to statistical normality.

Binomial Distributions and Normal Approximation

• Binomial Definition: Total successes in $n$ independent trials, each with probability $p$.
• Normal Approximation: For large n:
  • $\sum<em>{i=1}^{n}X</em>i \rightarrow N(np, \sqrt{np(1-p)})$.

Albino Example Summary

• Random Variables: Probabilities calculated using binomial distribution principles.
  • E.g., carriers of the albino gene producing children, applying normal approximation for large sample sizes to simplify computations.

Continuity Correction in Approximations

• Necessity: Correction employed when approximating a discrete binomial distribution with continuous normal distribution for more accurate probability assessments.

Summary of Findings

1. Sampling Distribution Properties:
   • Mean of sampling distribution $M_Y = µ$.
   • Standard deviation relationship: $σ_Y = \frac{σ}{\sqrt{n}}$.
   • Distribution shape:
     • If population distribution is normal, sampling distribution is normal (any sample size).
     • If $n$ is large, sampling distribution of $Y$ is approximately normal regardless of underlying population distribution.