Question

checkpoint question (2/2): sleep awareness week begins in the spring with the release of the national sleep foundations annual poll of u.s. sleep habits and ends with the beginning of daylight savings time, when most people lose an hour of sleep. in the foundations random sample of 1029 u.s. adults, 48% reported that they “often or always” got enough sleep during the past 7 nights. select the best answer regarding the conditions required for constructing a confidence interval for p: our unknown population proportion. 1029(.48)≥10 1029(.52)≥10 the randomization condition is met since a random sample of 1029 u.s. adults was selected. 1029 is less than 10% of all u.s. adults. all of the above statements are true.

Explanation:

Step1: Check the success - failure condition

For a confidence interval of proportion \(p\), the success - failure condition requires \(np\geq10\) and \(n(1 - p)\geq10\). Here \(n = 1029\), \(p=0.48\), \(n\times p=1029\times0.48 = 493.92\geq10\), and \(n\times(1 - p)=1029\times0.52=535.08\geq10\).

Step2: Check the randomization condition

The sample is a random sample of U.S. adults, so the randomization condition is met.

Step3: Check the 10% condition

Assume the population of U.S. adults is large. Since \(1029\) is likely less than \(10\%\) of all U.S. adults (if the population of U.S. adults is, say, more than \(10290\), which is very reasonable as the U.S. adult population is in the hundreds of millions).

Answer:

All of the above statements are true.