Introduction to Statistics - Statistical Inference

Introduction to Statistics - Statistical Inference

• The lecture was about statistical inference
• Learning objectives included understanding and defining key concepts, like sampling variation, sampling distributions, normal distributions, the Central Limit Theorem, standard error, standard deviation, confidence intervals, and p-values.
• The lectures assume prior knowledge of means, standard deviations (from descriptive statistics). Basic R functions are also assumed.
• Students should consult Chapters 3, 4, and 5 of the Navarro textbook for a refresher on these topics. (link provided)

British Social Attitudes Survey

• A British Social Survey from 2010-2021 showed that climate change is the most important environmental problem.
• Data regarding the survey's methodology and sample sizes was provided.

Sampling Variation

• Differences between samples and the population are random.
• This is called sampling variation
• Sampling bias is different from this variation (sampling bias occurs due to systematic forces like only interviewing people who answer their phones).
• This involves how best to estimate population values, given sampling variation.

Frequentist vs. Bayesian Statistics

• The lecture introduces frequentist statistics, which is the perspective of this course.
• The alternative approach is Bayesian statistics, which differs fundamentally.
• Frequentist statistics are more commonly used in practice for things like, p-values, significance, null hypotheses, and confidence intervals.
• Bayesian statistics involve things like Bayes Factors, priors, and credible intervals.

Normal Distribution

• The normal distribution (also known as the Gaussian distribution or bell curve) was covered.

• Key characteristics of the normal distribution, including:

  • Mean (μ)
  • Standard Deviation (σ)

• Key formulas regarding calculating standard deviation: Σ(Χί – μ) 2 /N

• Visual representations of normal distributions and probability densities were shown.

Sampling Distributions

• The sampling distribution of the mean refers to the distribution of means when repeatedly sampling from a population.
• The mean of the sampling distribution is the same as the population mean.
• The standard deviation of the sampling distribution is the standard error, which decreases with an increase in sample size.
• Sampling distributions approach a normal distribution as sample size increases (Central Limit Theorem).

Standard Error

• The standard deviation of the sampling distribution is called standard error
• Its value decreases as the sample size increases. This is a helpful measure of sampling error variation and precision

Confidence Intervals

• A 95% confidence interval means that 95% of repeated samples will contain the population mean within the upper and lower bounds.
• Confidence intervals are calculated using the mean of the sample and the standard error.
• The size of the confidence interval decreases with increasing sample size.

p-values

• p-values are used to test hypotheses in frequentist statistics.
• The null hypothesis assumes no effect or no difference.
• The p-value gives the probability of observing a sample result as extreme or more extreme, assuming the null hypothesis is true.
• Results with p-values less than the alpha level (e.g., 0.05) are typically considered statistically significant. A p-value in this range means the sample result is unlikely to have occurred if the null hypothesis were true.

One-Sample Example

• A one-sample example concerning 36 children who have completed a program is described.
• The population parameters for IQ tests are known: mean is 100 and standard deviation is 15.
• Calculation steps for finding a p-value are detailed.

Summary

• We collect samples to make inferences about the wider population
• Sampling variation is unavoidable but can be reduced with larger sample sizes.
• Frequentist statistics rely on the Central Limit Theorem which describes how sampling distributions are normally distributed.
• Confidence intervals and p-values can be calculated from the Central Limit Theorem.
• p-values are useful for hypothesis testing, comparing an observed outcome to an expected outcome under the null hypothesis.