Key Concepts in Inferential Statistics

6.1: Inferential Statistics

• Descriptive vs. Inferential Statistics:
  • Descriptive statistics summarize data we know (e.g., sample mean).
  • Inferential statistics estimate parameters of the population using sample data.
• Big Ideas in Inferential Statistics:
  • Estimation: Generating estimates from sample data.
  • Hypothesis testing: Making decisions about the population (not covered in this section).

6.2: Parameters vs Estimates

• Population Parameters vs Sample Estimates:
  • Population parameters (e.g., mean $A4, standard deviation $D$
    5$): true characteristics of the entire population.
  • Sample estimates (e.g., mean $ar{X}$, standard deviation $s$): derived from a subset of the population and used as best guesses.
• Example - IQ Scores:
  • True population mean: $D=100$, population standard deviation: $D=15$ (normal distribution).
  • Sample of 100 individuals could yield a different mean (e.g., $ar{X}=98.5$) and standard deviation (e.g., $s=15.9$).
  • Different samples can provide different estimates that could differ slightly from the population parameter.

6.3: The Law of Large Numbers

• Increasing Sample Size:
  • Larger samples yield estimates closer to true population parameters (e.g., $D=100$) and reduce sampling error.
• Law of Large Numbers:
  • States that as sample size increases, sample mean approaches the population mean.
  • This principle justifies that collecting more data yields better approximations of population parameters.

6.4: Sampling Distribution of the Mean

• Importance of Sample Size:
  • Small sample sizes lead to large variability in sample mean; larger sizes yield more accurate estimates.
• Sampling Distribution of the Mean:
  • Represents the distribution of sample means over multiple samples.
  • Determines how much variability exists in the sample means.
• Example Experiment:
  • Sampling $N = 5$ individuals produces various sample means with increased variability, while larger samples stabilize estimates.

6.5: Sampling Distributions Exist for Any Statistic!

• Any Statistic has a Sampling Distribution:
  • E.g., maximum IQ score can also be analyzed for its sampling distribution.
  • This demonstrates that any statistic computed on samples can showcase its own sampling distribution; whether means, standard deviations, etc.

6.6: The Central Limit Theorem

• Key Aspects of Central Limit Theorem (CLT):
  • As sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the population's shape.
• Sampling Distribution Properties:
  • The mean of sampling distribution = population mean $D$.
  • The standard deviation of sampling distribution (standard error, $ SEM $) decreases as sample size increases:
    SEM = \frac{\sigma}{\sqrt{N}} $$
• Standard Error Interpretation:
  • With larger sample sizes, expected differences from the population mean decrease (e.g., $ SEM $ for $ N=300 $ is $ 0.87 $, while for $ N=1000 $, it's $ 0.47 $).
  • This measure is crucial for statistical inference and estimating uncertainty in sample statistics.