(Solved): (7 points) I can see clearly, now According to a recent report by the World Health Organization \( ...

(7 points) I can see clearly, now According to a recent report by the World Health Organization \( ( \) WHO) at least \( 2.2 \) billion people around the world have a vision impairment. Stacey is a public health nurse that is concerned about access to eye care services in a low-income, rural area of the United States. As part of her investigations she obtains a random sample of 405 residents in this area and finds that 71 wear contact lenses to correct their vision. Round all calculated answers to 4 decimal places where appropriate. 1. Using the information from Stacey's sample, construct a \( 99 \% \) confidenoe interval for the true proportion of residents in this area who wear contact lenses. 2. Which of the following conditions must be met for the confidence interval to be valid? Select all that apply. A. The sample proportion must be normally distributed. B. The value for \( \rho \) must be less than \( 0.10 \) to provide evidence against the null hypothesis. C. The observations must be independent of one another. D. There must be at least 10 'success' and 10 failure' observations. 3. The information from Stacey's sample gives a standard error of \( S E_{\bar{p}}=0.0189 \). Which of the statements below is a correct interpretation of the standard error? A. The sample proportion is different from the true proportion of people who wear contact lenses approximately \( 1.89 \% \) of the time. B. On average, for repeated samples of this size, we expect the sample proportion to be approximately \( 0.0189 \) from the true population proportion. C. We have strong evidence that the true proportion of people who wear contact lenses is contained in a confidence interval. D. We can be \( 1.89 \% \) confident that our sample proportion is correctly calculated. 4. If Stacey wants to estimate the true proportion of residents in this area who wear contact lenses with \( 99 \% \) confidence and a margin of error of no more than \( 4.5 \% \), what size sample should she take? Use the (unrounded) information from Stacey's sample for a planning value for \( p^{*} \). Round your \( z^{*} \) value to exactly 3 decimal places. \( n= \) 5. Another question on Stacey's survey asks if respondents use corrective lenses while driving. From her data, Stacey calculates a \( 95 \% \) confidence interval for the proportion of residents who use corrective lenses while driving to be \( (0.553,0.625) \). Which of the following statements are appropriate interpretations in this scenario? Select all that apply. A. We can be \( 95 \% \) confident that, on average, the margin of error will vary no more than the size of the standard error. B. We can be \( 95 \% \) confident that the true proportion of residents who wear corrective lenses while driving will be contained in the interval \( 553,0.625) \). C. If Stacey colected 100 samples of size \( n=405 \) from this population and constructed 100 confidence intervals, she could expect approximately 95 of them to contain the true proportion of residents who wear contact lenses while driving. D. On average, \( 85 \% \) of the time we can expect any sample proportion from a sample of 405 residents to be in the interval \( (0.553,0.625) \). 6. If instead Stacey calculates a \( 90 \% \) confidence interval for the true proportion of residents who use corrective lenses while driving, this new interval would be the \( 95 \% \) confidence interval.