Lesson 10: Inference for Proportions

September 26, 2024

Review:

• Inference for a population mean
• Confidence Intervals
• Problem solutions

Presentation:

• How to Estimate a Population Proportion
  • Conduct a Simple Random Sample (SRS) of size n
  • Record the count, x, of some attribute attributed to a portion of the population (e.g., number of voters favoring a candidate)
  • Calculate the sample proportion, p-hat = x/n
  • If n is sufficiently large (≥30), we can assume p-hat is Normally distributed
  • Estimate of the population proportion mean, μ = p-hat
  • Estimate of the population proportion std dev, σ = √(p*(1-p)/n)
  • Estimate margin of error, m = z*σ  (use z = 1.96 for 95% confidence)
  • Estimate with 95% confidence interval is p-hat ± m
  • Same procedure for a small sample size (<30) using the t distribution and substituting t* for z*

• Example 8.1 on p. 489 – Binge Drinking Survey
  • n = 13,819
  • x = 3,140
  • p-hat = 3140/13819 = 0.227
  • standard deviation = √(p-hat*(1-p-hat)/n) = 0.00356
  • p-hat ± z*√(p-hat*(1-p-hat)/n) = 0.227 ± 1.96*(0.00356) = 0.227 ± 0.007
  • {0.220,0.234}
• Video – Against All Odds Unit 28

Activity:

1. A random sample of 2,454 12th-grade students were asked the following question: Taking all things together, how would you say things are these days – would you say you’re happy or not too happy? Of the responses, 2,098 students selected happy. Determine the sample proportion of students who responded they were happy and calculate a 95% confidence interval for the population proportion of 12th-grade students who are happy.
2. A phone survey contacted 1,910 households in which a computer was owned and respondents were asked if they could access the Internet from their home. A total of 1,816 of the households responded yes. Calculate a 95% confidence interval to estimate the proportion of American households with internet access.
3. Currently, mothers in North America are advised to put babies to sleep on their backs. This recommendation has reduced the number of cases of sudden infant death syndrome (SIDS). However, it is a likely cause of another problem, i.e., flat spots on babies’ heads. A study of 440 babies aged 7 – 12 weeks found that 46.6% had flat spots on their heads. Calculate a 95% confidence interval for the proportion of babies in this age group that have flat spots.

Assignment:

• Prepare for a quiz on Tue covering lessons 9-10 calculations