Sampling Error and Confidence Intervals

• Inferential Statistics


  A set of statistical procedures to test hypotheses (i.e., make inferences) about a population
• Very rarely are we only interested in only describing a sample, generally we are interested in making inferences from our sample to the population
• Inferential Statistics


  • Does this mean frog size represent all frogs in this lake?
  • Is anxiety of my sample related to sleeplessness in most adults?
  • Is the effect of my new pain medication as large as it was in my study?
• Sampling error


  Differences between a population parameter and sample statistic that is the result of the sampling procedure
• Sampling error is unknown in real life because we do not know the population parameter
• Statisticians became very concerned with Sampling Error
• Inferential Statistics


  1. Looked at ways to estimate sampling error from descriptive statistics in a sample

  2. Most of the procedures we use were designed to account for sampling error

  3. Confidence Intervals: give an estimate of sampling error

  4. NHST: give a probability that our results are due to sampling
• Probability Distribution


  A way to link a possible value of a random variable with the probability of occurrence

  
• Characteristics of probability distributions


  • Can be any shape
  • Represented by curve
  • Area under the curve represents the probability
  • Total area under the curve is always 1
  • Area under the curve between x-values represents the probability of getting those x values

  
• Probability Distributions


  • Uniform Distributions
  • Chi-Square

  
• Normal Distribution


  • Most concerned with normally distributed variables
  • Example: IQ scores with μ = 100, σ = 15

  
• Standard Normal Distribution


  A normal distribution with a mean of 0 and standard deviation of 1
• 1. score


  The number of standard deviations a value is away from the mean

  
• 1. scores give us a marker to calculate area under the curve

  
• Computers will give us exact probabilities from z-scores
• Sampling Distribution


  A distribution of a sample statistic (usually the mean) that would occur if we took an infinite number of samples from a population
• Characteristics of Sampling Distributions


  • Normal
  • M = μ
  • SD = σ/SQRT(N)

  
• Standard Error


  The average sampling error we can expect in our sample
• Confidence Interval


  An interval estimate of a population parameter from a sample statistic
• Steps to calculate a Confidence Interval


  1. Select your level of confidence

  2. Collect a sample from the population

  3. Calculate the mean of the sample

  4. Calculate the standard error (SD/N)

  5. Look up the critical value for your given level of confidence (z-score associated with the probability you want)

  6. Calculate the upper and lower bounds using the formula
• The mean was 425, 95% CI [422.47, 427.53]