Statistics Exam Notes

Level of Measurement

  • Level of measurement defines what operations and summaries are appropriate for data.
  • Nominal: categories with no intrinsic order; data are qualitative.
    • Examples: gender, eye color.
  • Ordinal: categories with a meaningful order but not equal intervals.
  • Interval: numeric scale with equal intervals, but no true zero.
  • Ratio: numeric scale with a true zero, allowing meaningful ratios.
  • In the transcript:
    • Data on Gender are nominal (level of measurement).
    • Eye color is Qualitative data (nominal).
    • Quantitative data refer to numbers and can be on interval or ratio scales depending on context.

Data Types

  • Qualitative (categorical) vs Quantitative (numeric).
  • Transcript examples:
    • Data collected as gender → Qualitative (categorical).
    • Eye color → Qualitative data (nominal).
    • Survey response: yes, no, undecided → Qualitative (categorical).
    • Data described as Quantitative data (numbers) → Numerical values.

Parameter vs Statistic

  • Parameter: numerical summary of the entire population.
  • Statistic: numerical summary of a sample.
  • Transcript examples:
    • a) Two thirds of the class are freshmen → Parameter
    • b) In a sample of students who passed statistics, 70% used statistics in their future careers → Statistic
    • c) Out of a sample of 1025 men, 85% like chocolate → Statistic
    • d) In a recent sample of 250 people, 25% do not bathe every day → Statistic

Discrete vs Continuous

  • The average weight of newborn babies in ounces → Continuous (can be measured to decimals).
  • Note:
    • Discrete data are counts (e.g., number of cars).
    • Continuous data are measurements that can take on an infinite number of values within an interval.

Observational vs Experimental

  • Q5: Does this describe an observational study or an experiment? The gender of children born in January were tallied → Observational Study (no manipulation or random assignment).

Experimental Design: HPV Vaccine Trial

  • Setup: A team tests the effectiveness of a new HPV vaccine by randomly dividing subjects into two groups.
    • Group 1 receives the new HPV vaccine (treatment group).
    • Group 2 receives the existing HPV vaccine (control group).
  • Blinding:
    • Participants were told which group they were in → Not blinded.
    • Therefore, not blind and not double-blind.
  • Correct description:
    • This is a Controlled Experiment, specifically a Randomized Controlled Trial (since subjects are randomly assigned and there is a control group).

Bias, Sampling, and Methods

  • Question 7: In a survey asking how many alcoholic drinks they consume each day, potential bias is Response bias (participants may not be honest).
    • Other related biases to know: Nonresponse bias (when individuals do not respond) and Sampling bias (systematic error due to sampling method).
  • Question 8: If the sample is chosen by asking our 40 closest friends, the sampling method is Convenience sampling (not random, potentially biased).

Frequency Distributions and Class Width

  • Data: 300 fish from the North Atlantic with lengths (mm) and frequencies:
    • 60-77: 1
    • 78-95: 16
    • 96-113: 71
    • 114-131: 108
    • 132-149: 83
    • 150-167: 18
    • 168-185: 3
  • Total frequency: 300 (checks out).
  • (a) Class width:
    • If classes are 60-77, 78-95, etc., the width is: w=U−L+1=77−60+1=18.w = U - L + 1 = 77 - 60 + 1 = 18.
  • (b) Class midpoint for the fifth class (132-149):
    • Midpoint: m=L+U2=132+1492=140.5.m = \frac{L+U}{2} = \frac{132+149}{2} = 140.5.

Quick Reference Formulas

  • Class width for equal-width classes: w=U−L+1.w = U - L + 1. (example gives 18 for these classes)
  • Class midpoint: m=L+U2m = \frac{L+U}{2}
  • Parameter vs Statistic: Parameter = population; Statistic = sample.
  • Data types: Qualitative vs Quantitative; Level of measurement informs appropriate analyses.