Probability: Sample Distributions

Sample Distribution

Probability

• A sample distribution is a probability distribution of a statistic obtained by selecting multiple samples from a population.
• The sample distribution is used to make inferences about the population parameter.
• Probability concepts:
  • Law of Large Numbers (LLN): the average of the results will converge to the population mean as the sample size increases.
  • Central Limit Theorem (CLT): the distribution of the sample mean will be approximately normal, even if the population distribution is not normal.

Random Sampling

• Random sampling is a method of selecting a sample from a population to ensure representativeness.
• Types of random sampling:
  • Simple Random Sampling: every individual in the population has an equal chance of being selected.
  • Stratified Random Sampling: the population is divided into subgroups, and random samples are selected from each subgroup.
  • Cluster Random Sampling: the population is divided into clusters, and random samples are selected from each cluster.

Confidence Intervals

• A confidence interval is a range of values within which the population parameter is likely to lie.
• Confidence level: the probability that the interval contains the population parameter (e.g., 95% confidence level means 95% of the intervals will contain the parameter).
• Margin of error: the maximum amount by which the sample statistic may differ from the population parameter.

Bias Correction

• Bias: a systematic error in the sample statistic that causes it to differ from the population parameter.
• Bias correction techniques:
  • Bootstrapping: resampling the data with replacement to estimate the bias.
  • Jackknifing: systematically removing one observation at a time to estimate the bias.

Hypothesis Testing

• A statistical test used to determine whether a hypothesis about the population parameter is true or not.
• Null hypothesis (H0): a statement of no effect or no difference.
• Alternative hypothesis (H1): a statement of an effect or a difference.
• Test statistic: a numerical value used to determine whether to reject or fail to reject the null hypothesis.
• P-value: the probability of observing the test statistic (or more extreme) assuming the null hypothesis is true.

Percentile

• A percentile is a value below which a certain percentage of the data falls.
• Types of percentiles:
  • 25th percentile (Q1): the value below which 25% of the data falls.
  • 50th percentile (median): the value below which 50% of the data falls.
  • 75th percentile (Q3): the value below which 75% of the data falls.

Sample Distribution

• A sample distribution is a probability distribution of a statistic obtained by selecting multiple samples from a population.
• It's used to make inferences about the population parameter.
• Key probability concepts include:
  • Law of Large Numbers (LLN): the average of the results will converge to the population mean as the sample size increases.
  • Central Limit Theorem (CLT): the distribution of the sample mean will be approximately normal, even if the population distribution is not normal.

Random Sampling

• Random sampling is a method of selecting a sample from a population to ensure representativeness.
• Types of random sampling include:
  • Simple Random Sampling: every individual in the population has an equal chance of being selected.
  • Stratified Random Sampling: the population is divided into subgroups, and random samples are selected from each subgroup.
  • Cluster Random Sampling: the population is divided into clusters, and random samples are selected from each cluster.

Confidence Intervals

• A confidence interval is a range of values within which the population parameter is likely to lie.
• Confidence level: the probability that the interval contains the population parameter (e.g., 95% confidence level means 95% of the intervals will contain the parameter).
• Margin of error: the maximum amount by which the sample statistic may differ from the population parameter.

Bias Correction

• Bias: a systematic error in the sample statistic that causes it to differ from the population parameter.
• Bias correction techniques include:
  • Bootstrapping: resampling the data with replacement to estimate the bias.
  • Jackknifing: systematically removing one observation at a time to estimate the bias.

Hypothesis Testing

• A statistical test used to determine whether a hypothesis about the population parameter is true or not.
• Null hypothesis (H0): a statement of no effect or no difference.
• Alternative hypothesis (H1): a statement of an effect or a difference.
• Test statistic: a numerical value used to determine whether to reject or fail to reject the null hypothesis.
• P-value: the probability of observing the test statistic (or more extreme) assuming the null hypothesis is true.

Percentiles

• A percentile is a value below which a certain percentage of the data falls.
• Types of percentiles include:
  • 25th percentile (Q1): the value below which 25% of the data falls.
  • 50th percentile (median): the value below which 50% of the data falls.
  • 75th percentile (Q3): the value below which 75% of the data falls.