Chapter 6 - Inference for Categorical Data

6.5 Prop 19 in California.

In a 2010 Survey USA poll, \(70\%\) of the \(119\) respondents between the ages of \(18\) and \(34\) said they would vote in the \(2010\) general election for Prop \(19\), which would change California law to legalize marijuana and allow it to be regulated and taxed. At a \(95\%\) confidence level, this sample has an \(8\%\) margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning.

Answer:

(a) We are \(95\%\) confident that between \(62\%\) and \(78\%\) of the California voters in this sample support Prop \(19\).

False: As we know, a confidence interval is constructed to estimate the population proportion, not the sample proportion as it is stated.

(b) We are \(95\%\) confident that between \(62\%\) and \(78\%\) of all California voters between the ages of \(18\) and \(34\) support Prop \(19\).

True: As we know, a confidence interval is constructed to estimate the population proportion and from the above statement we have as follows:

\(95\%\) CI: \(70\% ± 8\%\); which means \(95\%\) CI goes from \(70\% - 8\% = 62\%\) to \(70\% + 8\% = 78\%\).

(c) If we considered many random samples of \(119\) California voters between the ages of \(18\) and \(34\), and we calculated \(95\%\) confidence intervals for each, \(95\%\) of them will include the true population proportion of \(18-34\) year old Californians who support Prop \(19\).

True: By the definition of the confidence level.

That is: Constructing a confidence interval for a proportion:

  • Verify the observations are independent and also verify the success-failure condition using \(\hat{p}\) and \(n\).

  • If the conditions are met, the sampling distribution of \(\hat{p}\) may be well approximated by the normal model.

  • Construct the standard error using \(\hat{p}\) in place of \(p\) and apply the general confidence interval formula.

(d) In order to decrease the margin of error to \(4\%\), we would need to quadruple (multiply by \(4\)) the sample size.

True: Quadrupling the sample size decreases the SE and ME by a factor of \(\frac{1}{\sqrt{4}}=\frac{1}{2}\).

(e) Based on this confidence interval, there is sufficient evidence to conclude that a majority of California voters between the ages of \(18\) and \(34\) support Prop \(19\).

True: The \(95\%\) CI is entirely above \(50\%\).