Question

1. modeling real life in a survey of 920 u.s. teenagers, 81% said that helping others who are in need will be very important to them as adults. give an interval that is likely to contain the exact percentage of all u.s. teenagers who think that helping others who are in need will be very important to them as adults. round your answers to the nearest tenth. the interval is between 1 % and 2 %. correct answers: 20. modeling real life in a survey of 2000 u.s. adults, 70% said that they would support a national policy requiring rooftop solar panels to be installed on all new homes. give an interval that is likely to contain the exact percent of all u.s. adults who would support this policy. round your answers to the nearest tenth. the interval is between 1 % and 2 %. correct answers: 21. repeated reasoning what happens to the margin of error as a sample size increases? the margin of error 1.

Explanation:

Step1: Recall margin - of - error formula for proportion

For a proportion $p$ in a sample of size $n$, the margin of error $E$ at a 95% confidence level is given by $E = 1/\sqrt{n}$ (a common approximation).

Step2: Solve for problem 19

Given $n = 920$ and $\hat{p}=0.81$. The margin of error $E = 1/\sqrt{920}\approx1/\ 30.33\approx0.033$. The confidence - interval is $\hat{p}-E

Step3: Solve for problem 20

Given $n = 2000$ and $\hat{p}=0.70$. The margin of error $E = 1/\sqrt{2000}\approx1/44.72\approx0.022$. The confidence - interval is $\hat{p}-E

Step4: Analyze margin of error and sample size relationship

The formula for margin of error $E$ for a proportion is $E = z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ (where $z$ is the z - score corresponding to the desired confidence level). As $n$ increases, the denominator of the fraction $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ increases, so the value of $E$ decreases.

Answer:

1. The interval is between $77.7\%$ and $84.3\%$.
2. The interval is between $67.8\%$ and $72.2\%$.
3. The margin of error decreases.