A newsgroup is interested in constructing a $95\%$ confidence interval for the difference in the proportions of Texans and New Yorkers who favor a new Green initiative. Of the $534$ randomly selected Texans surveyed, $405$ were in favor of the initiative and of the $561$ randomly selected New Yorkers surveyed, $457$ were in favor of the initiative.

a$.$ With $95\%$ confidence the difference in the proportions of Texans and New Yorkers who favor a new Green initiative is between $($round to $3$ decimal places$)$ and $($round to $3$ decimal places$).$

b$.$ If many groups of $534$ randomly selected Texans and $561$ randomly selected New Yorkers were surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of the difference in the proportions of Texans and New Yorkers who favor a new Green initiative and about percent will not contain the true population difference in proportions.