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• Margin of error

  A statistic expressing the amount of allowable random sampling error in results
• Making a Decision

  Deciding on the alpha depends on how much tolerance you have for a false positive
• Three common confidence levels: 90%, 95%, 99%
• Testing the Difference Between Two Means


  Suppose a researcher wishes to determine whether there is a difference in the average age of students enrolled at different colleges
• Hypothesis Testing

  1. Every hypothesis-testing situation begins with the statement of a hypothesis

  2. Two types of statistical hypothesis for each situation
• Null hypothesis

  Can be rejected based on statistical evidence
• T Test for Two Means


  1. Definition: A parametric test that is usually used to measure significant difference of small sample sizes

  2. Two types: Dependent t-test and Independent t-test
• Situations for hypothesis testing

  • Situation A: Medical researcher interested in side effects of a new medication
  • Situation B: Medical technologist invents a new diagnostic test kit
  • Situation C: Chief pathologist wishes to decrease the cost of kidney function tests
• Sample Size Determination

  1. The act of choosing the number of observations or replicates to include in a statistical sample

  2. Goal: used in any empirical study to make inferences about a population from a sample
• Alternative hypothesis


  Can be used to support a claim
• Confidence level critical values

  • 90%: +1.65
  • 95%: +1.96
  • 99%: +2.58
• Confidence Intervals

  1. Suppose a college president wishes to estimate the average age of students attending this semester. The president could select a random sample of 100 students and find the average age of these students, say 22.3 years.

  2. If the sample size is >30, the distribution of the means will be approximately normal
• Z Test for a Mean


  1. If σ is KNOWN and n > 30, use the z test

  2. If σ is KNOWN, and n < 30, use the t test

  3. If σ is UNKNOWN, but n > 30, use the t test

  4. If σ is UNKNOWN, and n < 30, use the t test
• 1. Value Method

  Interpreting P-Values
• F Test


  Comparison of two sample/population variances or standard deviations which should be independent from each other
• Scheffe test


  • Can be used when samples are of different sizes
  • Can be used to make pairwise comparisons (sample 1 + sample 2 VS sample 3)
• Correlation


  Determines whether there exists a relationship between variables
• ANOVA Interpretation


  • No differences in the means: between-group = within-group, F test value approx. equal to 1
  • Means differ significantly: between-group > within-group, F test will be significantly >1, Significant test value: A high probability that the difference is not due to chance alone, F test to compare means is always right-tailed (+)
• Scatter Plots

  Pearson Product Moment Correlation Coefficient
• Post-Hoc Tests

  • Scheffé / Scheffé’s test
  • Tukey test / Tukey HSD (honestly significant difference)
• Non-parametric tests are sometimes called "distribution-free" tests as they have fewer assumptions of the data
• Learning Outcomes

  1. Use the one-way ANOVA technique to determine if there is a significant difference among three or more means

  2. Determine which means differ, using Post-Hoc Tests (Scheffé or Tukey HSD) when an ANOVA is significant (rejected H0)

  3. Use the two-way ANOVA technique to determine if there is a significant difference in the main effects or interaction

  4. Determine the correlation of data sets and test hypothesis

  5. Understand the purpose of scatter plots and regression lines

  6. Identify non-parametric tests equivalent to parametric tests

  7. Test variances and standard deviations, using chi-square test
• Post-Hoc Tests Procedure

  Prepare the means two at a time, using all possible combination of means (pair-wise combinations)
• Analysis of Variance

  • When an F test is used to test a hypothesis concerning the means of 3 or more populations
  • Sample sizes need not to be equal
  • Two estimates: Between-group variance – finding variance of the means, Within-group variance – finding the variance for all data
• Regression


  Describe the nature of the relationship between variables
• Application in the Lab

  • One-way ANOVA: Independent - Soil type, Dependent - Plant growth
  • Two-way ANOVA: Independent - Soil type, Plant food, Dependent - Plant growth
• Non-Parametric Tests

  • Chi-Square Distribution
• Tukey test


  • When samples are equal in size
  • Narrower confidence limit
  • More powerful than the Scheffe test for making pairwise comparisons for the mean
• Non-parametric tests are used when the variables are ordinal or nominal or when there are definite outliers or detection limits