Class 12

Week 7: Biostatistics Overview

1. Introduction to Inferential Statistics

  • Purpose: Inferential statistics is used to draw conclusions about a population based on sample data.

  • Key Concepts:

    • Population: Complete set being studied.

    • Sample: Subset of the population used for analysis.

    • Parameter: A descriptive measure of a population (e.g., population mean, population standard deviation).

    • Statistic: A descriptive measure of a sample (e.g., sample mean, sample standard deviation).

2. Sampling Distributions

  • Definition: The sampling distribution is the distribution of a statistic (e.g., mean) over all possible samples of a given size from a population.

  • Random Samples:

    1. Obtain a simple random sample of size n.

    2. Calculate the sample mean.

    3. Repeat until all possible simple random samples of size n are obtained.

  • Characteristics:

    • Each sample has its own mean.

    • Collectively, these sample means form the sampling distribution of the sample mean.

3. Central Limit Theorem (CLT)

  • Importance: As sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the population’s distribution shape.

  • Conditions for Normality:

    • If the population distribution is normal, the sampling distribution of the mean will also be normal for any sample size N.

    • If the population is not normal, a sample size of about 30 (N=30) is generally sufficient for the sampling distribution to appear normal.

    • With highly skewed populations or distributions with extreme outliers, larger samples may be necessary (N=500 or more) to achieve normality in the sampling distribution.

4. Standard Error

  • Definition: Standard error (SE) measures the dispersion of sample means around the population mean.

    • Formula for SE of the Mean:

      • SE = σ / √n, where σ is the population standard deviation and n is the sample size.

  • Relationship to Population Standard Deviation: The standard error is typically smaller than the population standard deviation due to reduced variability with summary measures compared to individual measurements.

5. Examples of Sampling Distributions

  • If the population distribution is significantly non-normal, the distribution of the means will still tend to be normal with an increase in sample size.

  • Examine distributions of various population shapes and their corresponding sampling distributions:

    • Normal population distribution yields a normal sampling distribution.

    • Non-normal populations (skewed, bimodal, etc.) show normality in the sampling distribution as sample size increases.

6. Application Example: Latency Times to AIDS

  • Given: Mean latency time = 5.24 years, std deviation = 10.15 years.

    • Compute probability that a randomly sampled individual has latency time ≥ 8 years using N distribution.

    • For a sample of size 50, compute the probability that the mean latency time is ≥ 8 years using the standard error formula:

      • SE = σ / √n = 10.15 / √50 = 1.44.