## Lesson 16: Inference for Proportions

October 29, 2018

Review:

- t-distribution for small samples
- class on Thu

Presentation:

- How to Estimate a Population Proportion
- Conduct a Simple Random Sample (SRS) of size
*n* - Record the count,
*x*, of “successes”, e.g., # of voters supporting candidate A - 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*

- Conduct a Simple Random Sample (SRS) of size
- 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}

Activity:

- 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.
- 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.
- 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.

Assignment:

You will be assigned a number of registered voters in Congressional District 3. You will telephone each voter on the list and ask two questions. First, introduce yourself as a CSU-Pueblo student working on an assignment for your statistics class. Then, ask these two questions:

- Do you plan to vote in next week’s election?
- Who do you support in the race for US House of Representatives: Republican Scott Tipton or Democrat Diane Mitsch Bush?

Ask both questions, even if the answer to #1 is “no”. Once all surveys have been completed and response data compiled you will use the data to estimate vote percentages for both candidates in your assigned county.