Chapter 10 Two-Sample Confidence Intervals

1. Section 10.1 Confidence Intervals for the Difference Between Two Means: Independent Samples
   • Distinguish between independent samples and paired samples
   • Construct Confidence Intervals for the difference between two Population Means
   • point estimate for `mu_1 - mu_2` is `barx_1 - barx_2`
   • If the samples are independent then the variance of the difference `barx_1 - barx_2` is the sum of the variances.
   • If two variances are known, then
     variance of `barx_1 - barx_2 = (sigma_1^2)/(n_1) + (sigma_2^2)/(n_2)`
   • If the variances are not known then we approximate them by the sample standard deviation
   • The standard error of the difference of the means is the same as the standard deviation of the difference of the means, i.e., the square root of the variance,
     standard error of `barx_1 - barx_2 = sqrt((s_1^2)/(n_1) + (s_2^2)/(n_2))`
   • Margin of error = `t_(alpha/2)*sqrt((s_1^2)/(n_1) + (s_2^2)/(n_2))`
   • Confidence interval given by

     `barx_1 - barx_2 +- t_(alpha/2)*sqrt((s_1^2)/(n_1) + (s_2^2)/(n_2))` using the smaller of `(n_1-1)` and `(n_2-1)` as the degrees of freedom.

   • The more complicated formula for degrees of freedom:
     degrees of freedom = `(( (s_1^2)/(n_1) + (s_2^2)/(n_2) )^2) / ((((s_1^2)/(n_1))^2)/(n_1-1)+(((s_2^2)/(n_2))^2)/(n_2-1))`
   • When it is known that `sigma_1 = sigma_2` but we do not know their value, then we can use the pooled standard deviation method,

     ` s_p = sqrt( ( (n_1 -1)*s_1^2 +(n_2-1)*s_2^2)/ (n_1+n_2 - 2) )`

     but the book advises against it. (Technology is much better.)

   • If we actually know `sigma_1` and `sigma_2` then

     `barx_1 - barx_2 +- z_(alpha/2)*sqrt((sigma_1^2)/(n_1) + (sigma_2^2)/(n_2))`

2. Section 10.2 Confidence Intervals for the Difference Between Two Proportions
   • point estimate for `p_1 - p_2` is `hatp_1 - hatp_2`
   • Variance of `hatp_1 = ( hatp_1*(1-hatp_1) )/(n_1)` and Variance of `hatp_2 = ( hatp_2*(1-hatp_2) )/(n_2)`
   • Variance of `hatp_1 - hatp_2 = ( hatp_1*(1-hatp_1) )/(n_1) + ( hatp_2*(1-hatp_2) )/(n_2)`
   • Standard error of `hatp_1 - hatp_2 = sqrt(( hatp_1*(1-hatp_1) )/(n_1) + ( hatp_2*(1-hatp_2) )/(n_2) )`
   • Margin of Error = `(z_(alpha/2))* sqrt(( hatp_1*(1-hatp_1) )/(n_1) + ( hatp_2*(1-hatp_2) )/(n_2) )`
   • Confidence Interval is `hatp_1 - hatp_2 +- (z_(alpha/2))* sqrt(( hatp_1*(1-hatp_1) )/(n_1) + ( hatp_2*(1-hatp_2) )/(n_2) )`
3. Section 10.3 Confidence Intervals for the Difference Between Two Means: Paired Samples
   • Assumptions and Notation
     • Two paired samples
     • Sample size is large (n>30), or underlying population is approximately normal
     • `bard` is the sample mean of the differences in the matched pairs
     • `s_d` is the sample standards deviation of the differences in the matched pairs
     • `mu_d` is the population difference for the matched pairs
   • Confidence interval is:

     `bar d +- t_(alpha/2) *( s_d/sqrt(n) )`

     `bar d - t_(alpha/2) *( s_d/sqrt(n) ) < mu_d < bar d + t_(alpha/2) *( s_d/sqrt(n) )`

4. Sample problems on the calculator.