Inferential Statistics

• Allow Generalizations about Populations Using Data from Samples
• Inductive Reasoning – reasoning from the particular to the general and from the observed to the unobserved
• Sample – subset (n) of a population (N) selected so that the characteristics of the population can be estimated with a known margin of error
• Representativeness – the size of the sample is less important than that the sample accurately represent the population
• Random Sample – "convenience" samples are of little value in estimating parameters for a population
• Independence – regardless of whatever other subjects were chosen, all other subjects must have an equal probability of being chosen
• Parameters – descriptive measures computed for populations
• Statistics – descriptive measures computed for samples
  

• Hypothesis - a statement about one or more parameters
• Scientific Hypothesis – a suggested solution to a problem (educated guess)
• Statistical Hypothesis – (H_o) a numerical statement about an unknown parameter – a decision as to whether or not H_o is false 
• Null Hypothesis – the hypothesis of no difference or no relationship 
• Errors in Hypothesis Testing
  

• Testing Hypotheses About 
• State the statistical hypothesis Ho to be tested
• Specify the degree of risk of type-I error (incorrectly concluding H_o is false) stated as a probability  (alpha level of significance) of a type-I error (e.g., =.05)
• Assuming H_o to be correct, determine the probability (p) of obtaining a sample mean that differs from  by an amount as large or larger than that which was observed
• Decide whether or not to reject H_o (e.g., if probability (p) is less than , H_o is rejected)

• The values of a statistic computed from all possible samples of a given size (n)
• For each valid sample:
• Sample mean reflects (approximates) the population mean
• Sample mean not expected to equal population mean due to sampling fluctuations
  
• Central Limit Theorem
• Even for non-normal parent populations, the shape of the sampling distribution rapidly approaches normal as n increases
• As n increases, the variability of the sampling distribution of the means decreases even if the parent population is non-normal
• Standard Error of the Mean (standard deviation of the sampling distribution) computed as:
  
• Applicable for non-normal populations if: n ³ 30

• An interval estimation of a parameter
• Symmetric around the mean
• Confidence Coefficient – likelihood that parameter  will be contained within the confidence interval
• Typically, 95% (19 out of 20)