Lecture Notes: Behavior of Sample Proportions

Key Questions

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

Context

• Focus on the population of part-time college students
• Assumed population proportion: 60% female
• Random samples of 25 students are taken

Sampling Process

• Sample 1: 17 out of 25 females, sample proportion (p-hat) = 0.68
• Sample 2: 18 out of 25 females, sample proportion = 0.72
• Sample 3: 16 out of 25 females, sample proportion = 0.64
• Observation: Each sample yields a different proportion of females, indicating variability

Investigation

• Objective: Understand what happens when many random samples are collected
• Conducted over 2,000 random samples
  • Each sample consists of 25 part-time college students
  • Calculated and recorded sample proportions for each

Results

• Many samples had proportions close to the population proportion of 0.6
• Fewer samples had extreme proportions (far from 0.6)
• Standard Deviation: Approximately 10%
  • Indicates typical sample proportions between 0.5 and 0.7

Graphical Analysis

• Shape of sampling distribution is approximately normal
• A normal distribution models the sample proportions well
• A normal model is a good probability model for sampling distributions

Next Steps

• Further investigation on variability in sample proportions
• Focus on the impact of sample size

Conclusion

• Normal distribution is a reliable model for the sampling distribution of sample proportions
• Future exploration will delve deeper into sample size effects on variability