Questions: Mastering Concepts and Skills 1. Satisfaction A recent Gallup Poll asked a random sample of 1025 adults, "In general, are you satisfied or dissatisfied with the way things are going in the United States at this time?" In all, 25% said that they were satisfied. The dotplot shows the proportion who said they were satisfied in each of 500 random samples of size 1025 from a population where 25% are satisfied. Distribution of simulated proportion # samples mean SD 500 0.25 0.014 (a) Use the results of the simulation to approximate the margin of error for Gallup's estimate of the proportion of U.S. adults who were satisfied with the way things were going in the United States at the time of the poll. (b) Interpret the margin of error.

Mastering Concepts and Skills 1. Satisfaction A recent Gallup Poll asked a random sample of 1025 adults, "In general, are you satisfied or dissatisfied with the way things are going in the United States at this time?" In all, 25% said that they were satisfied. The dotplot shows the proportion who said they were satisfied in each of 500 random samples of size 1025 from a population where 25% are satisfied. Distribution of simulated proportion # samples mean SD 500 0.25 0.014 (a) Use the results of the simulation to approximate the margin of error for Gallup's estimate of the proportion of U.S. adults who were satisfied with the way things were going in the United States at the time of the poll. (b) Interpret the margin of error.
Transcript text: Mastering Concepts and Skills 1. Satisfaction A recent Gallup Poll asked a random pg 2 sample of 1025 adults, "In general, are you satisfied or dissatisfied with the way things are going in the United States at this time?" In all, $25 \%$ said that they were satisfied. The dotplot shows the proportion who said they were satisfied in each of 500 random samples of size 1025 from a population where $25 \%$ are satisfied. \begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{ Distribution of simulated proportion } \\ \hline \# samples & mean & SD \\ \hline 500 & 0.25 & 0.014 \\ \hline \end{tabular} (a) Use the results of the simulation to approximate the margin of error for Gallup's estimate of the proportion of U.S. adults who were satisfied with the way things were going in the United States at the time of the poll. (b) Interpret the margin of error.
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Solution

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Solution Steps

Step 1: Find the range of plausible values.

The distribution is roughly symmetric and centered at 0.25. Almost all the values are between 0.22 and 0.28.

Step 2: Calculate the margin of error.

The margin of error is approximately half the range of plausible values. In this case, it's (0.28-0.22)/2 = 0.03.

Final Answer

(a) The margin of error is approximately 0.03. (b) We are reasonably sure that Gallup's estimate of 0.25 is within 0.03 of the true population proportion of U.S. adults who were satisfied with the way things were going in the United States at the time of the poll.

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