Chapter 8 Confidence Intervals

1. Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known
   • point estimate
   • confidence interval and the associated confidence level
   • Point estimate ± Margin of error
   • When `sigma` is known,

     `bar x +- z_(alpha/2) *( sigma/sqrt(n) )`

     `bar x - z_(alpha/2) *( sigma/sqrt(n) ) < mu < bar x + z_(alpha/2) *( sigma/sqrt(n) )`

   • The trade-off: For a given sample size, higher confidence level means a larger margin of error.
   • Finding the required sample size

     we can decrease the margin of error by increasing the sample size `n`.

     `n = ((z_(alpha/2)*sigma)/m)^2`

   • Interpreting a confidence interval
2. Section 8.2 Confidence Intervals for a Population Mean, Standard Deviation Unknown
   • Student's t
   • Degrees of freedom (n-1)
   • Look at table images on the web and at both the standard table and the special tables on this site.
   • Point estimate ± Margin of error
   • When `sigma` is unknown, use `s` and Student's t

     `bar x +- t_(alpha/2) *( s/sqrt(n) )`

     `bar x - t_(alpha/2) *( s/sqrt(n) ) < mu < bar x + t_(alpha/2) *( s/sqrt(n) )`

   • This works well when the sample size is `> 30`. When `n<=30` then we also want to be relatively sure that the underlying population has an approximately normal distribution.
   • For examples of confidence intervals see Looking at Confidence Intervals.
3. Section 8.3 Confidence Intervals for a Population Proportion
   • Assumptions for constructing a conidence interval for `p`
     • simple random sample
     • The population is at least 20 times as large as the sample (we sample less than 5% of the population)
     • Items in the population are in two categories
     • The sample must have at least 10 items in each category.
   • Point estimate ± Margin of error
   • `hat p +- z_(alpha/2) *sqrt( ((hat p)*(1-hat p))/n) `

     `hat p - z_(alpha/2) * sqrt( ((hat p)*(1-hat p))/n) < p < hat p + z_(alpha/2) * sqrt( ((hat p)*(1-hat p))/n)`

   • Finding the required sample size

     we can decrease the margin of error by increasing the sample size `n`.

     `n = hatp*(1-hatp)*((z_(alpha/2))/m)^2`

4. Sample problems.