Lecture Notes: Behavior of Sample Proportions

Key Questions

• What patterns emerge when collecting random samples?
• What is the shape, center, and spread of the distribution of sample proportions?

Context

• Examination of part-time college students population.
• Assumption: 60% of this population is female.

Investigation Process

• Collect random samples of 25 students.
• Calculate the sample proportion (( \hat{p} )) of females.
• Graph each ( \hat{p} ) to create a sampling distribution of sample proportions.

Sample Collection Examples

• First Sample:
  • 17 out of 25 students were female.
  • ( \hat{p} = 0.68 )
• Second Sample:
  • 18 out of 25 students were female.
  • ( \hat{p} = 0.72 )
• Third Sample:
  • 16 out of 25 students were female.
  • ( \hat{p} = 0.64 )

Observations

• Each random sample shows variability in the proportion of females.
• Variability is expected in random sampling.

Predictions & Outcomes

• Predictions: Consider the shape, center, and spread of the distribution when many samples are collected.
• Outcome of Simulation:
  • 2000 random samples collected.
  • Many sample proportions close to population proportion (0.6).
  • Fewer samples as the proportion moves further from 0.6.

Analysis of Distribution

• Standard Deviation:
  • Approximately 10% (0.10).
  • Typical sample proportions between 0.5 and 0.7.
• Shape:
  • Approximately normal.
  • Normal distribution models the sample proportions well.

Conclusions

• Normal model is a good probability model for sampling distribution of sample proportions.
• Next steps will involve investigating impact of sample size on variability.

Prepare for the next lecture where the focus will be on variability in sample proportions and the impact of sample size.