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: $w = U - L + 1 = 77 - 60 + 1 = 18.$
• (b) Class midpoint for the fifth class (132-149):
  • Midpoint: $m = \frac{L+U}{2} = \frac{132+149}{2} = 140.5.$

Quick Reference Formulas

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