A poll surveyed a random sample of 50 adults. The survey asked the following question: "Do you feel things are going well in the United States these days?" (Assume the only answers are Yes or No.) The results are in the table below.

No Yes No No Yes Yes No Yes No Yes

Yes No Yes No Yes No Yes No Yes No

Yes No Yes No Yes No Yes No Yes No

Yes No Yes No Yes No Yes No Yes No

No No No Yes No No No Yes Yes No

Calculate the $95\%$ confidence interval for the population proportion of those that said yes. Round your answers to three decimal places.

Question

02-11-2026
Mathematics

Answered

A poll surveyed a random sample of 50 adults. The survey asked the following question: "Do you feel things are going well in the United States these days?" (Assume the only answers are Yes or No.) The results are in the table below.

No Yes No No Yes Yes No Yes No Yes

Yes No Yes No Yes No Yes No Yes No

Yes No Yes No Yes No Yes No Yes No

Yes No Yes No Yes No Yes No Yes No

No No No Yes No No No Yes Yes No

Calculate the $95\%$ confidence interval for the population proportion of those that said yes. Round your answers to three decimal places.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T02:11:36-05:00

Count 'Yes' responses to find the sample proportion: $\hat{p} = \frac{23}{50} = 0.46$.
Calculate the standard error: $SE = \sqrt{\frac{0.46(0.54)}{50}} \approx 0.0705$.
Determine the margin of error at a 95% confidence level: $ME = 1.96 \times 0.0705 \approx 0.1382$.
Construct and round the confidence interval to three decimal places: $\boxed{0.322 < p < 0.598}$.

Explanation

Determine the sample proportion of 'Yes' responses

To determine the 95% confidence interval for the proportion of adults who answered 'Yes', we first count the number of 'Yes' responses from the provided survey data. By carefully reviewing each entry in the table, we identify a total of $23$ 'Yes' responses out of a sample size of $n = 50$ adults. Thus, the sample proportion is calculated as:
\[\hat{p} = \frac{23}{50} = 0.46\]

Calculate the standard error and margin of error

The formula for a 95% confidence interval for a population proportion is $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. For a 95% confidence level, the standard normal critical value is $z^* \approx 1.96$. First, we calculate the standard error (SE):
\[SE = \sqrt{\frac{0.46(1 - 0.46)}{50}} = \sqrt{\frac{0.46 \cdot 0.54}{50}} = \sqrt{\frac{0.2484}{50}} \approx 0.07048\]
Next, we calculate the margin of error (ME) by multiplying the critical value by the standard error:
\[ME = 1.96 \cdot 0.07048 \approx 0.13815\]

Construct the 95% confidence interval

Now, we construct the confidence interval by subtracting and adding the margin of error to the sample proportion:
\[\text{Lower Bound} = 0.46 - 0.13815 = 0.32185\]
\[\text{Upper Bound} = 0.46 + 0.13815 = 0.59815\]
Rounding these bounds to three decimal places, we obtain the interval $(0.322, 0.598)$. Thus, the 95% confidence interval for the population proportion is $0.322 < p < 0.598$.

Examples

Confidence intervals are essential tools in statistics for estimating population parameters based on sample data. For example, if a company wants to estimate the percentage of customers satisfied with their service, they can survey a random sample and construct a 95% confidence interval. This tells them, with 95% confidence, that the true proportion of satisfied customers lies between two bounds, allowing for better business decision-making. If $\hat{p} = 0.46$ and $n=50$, the margin of error is $z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \approx 0.138$, leading to an interval of $(0.322, 0.598)$.

No	Yes	No	No	Yes	Yes	No	Yes	No	Yes
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
No	No	No	Yes	No	No	No	Yes	Yes	No

No	Yes	No	No	Yes	Yes	No	Yes	No	Yes
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
No	No	No	Yes	No	No	No	Yes	Yes	No

No	Yes	No	No	Yes	Yes	No	Yes	No	Yes
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
Yes	No	Yes	No	Yes	No	Yes	No	Yes	No
No	No	No	Yes	No	No	No	Yes	Yes	No