A survey of households in a small town showed that in 500 of 1,200 sampled households, at least one member attended a town meeting during the year. Using the 95% level of confidence, what is the confidence interval for the proportion of households represented at a town meeting?

Phat = 500/1200

phat =0.4167
E = 1.96 * sqrt(.4167*.5833/1200)
E = 0.0279
0.4167 -+ 0.0279
[0.3888, 0.4446]

To calculate the confidence interval for the proportion of households represented at a town meeting, we can use the following formula:

CI = p̂ ± Z * √(p̂(1-p̂)/n)

Where:

- p̂ is the sample proportion (500/1200 = 0.4167)
- Z is the Z-score corresponding to the desired confidence level (95% level corresponds to a Z-score of 1.96)
- n is the sample size (1200)

Let's plug in the values:

CI = 0.4167 ± 1.96 * √(0.4167(1-0.4167)/1200)

Calculating the values within the square root:

0.4167(1-0.4167) = 0.2467
√(0.2467/1200) ≈ 0.0154

Plugging in the values into the formula:

CI = 0.4167 ± 1.96 * 0.0154

Now, we can calculate the confidence interval:

CI = (0.4167 - 1.96 * 0.0154, 0.4167 + 1.96 * 0.0154)

Simplifying and calculating:

CI = (0.4167 - 0.0301, 0.4167 + 0.0301)

CI = (0.3866, 0.4468)

Therefore, the confidence interval for the proportion of households represented at a town meeting is (0.3866, 0.4468) at a 95% level of confidence.

To calculate the confidence interval for the proportion of households represented at a town meeting, we can use the formula:

Confidence Interval = Sample Proportion ± Margin of Error

First, we need to find the sample proportion, which is the number of households represented at a town meeting divided by the total number of sampled households:

Sample Proportion = Number of households represented at a town meeting / Total number of sampled households

Sample Proportion = 500 / 1200 = 0.4167 (rounded to 4 decimal places)

Next, we need to find the margin of error. The margin of error depends on the level of confidence and the sample proportion. For a 95% level of confidence, we can use the following formula to find the margin of error:

Margin of Error = Z * sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)

In this formula, Z is the z-score associated with a 95% confidence level. For a two-tailed test, the z-score is approximately 1.96.

Sample Size = Total number of sampled households = 1200

Plugging in the values, we get:

Margin of Error = 1.96 * sqrt((0.4167 * (1 - 0.4167)) / 1200)

Calculating this, we find that the margin of error is approximately 0.0254 (rounded to 4 decimal places).

Now we can calculate the confidence interval:

Confidence Interval = Sample Proportion ± Margin of Error

Confidence Interval = 0.4167 ± 0.0254
Confidence Interval = (0.3913, 0.4421)

Therefore, at a 95% level of confidence, the confidence interval for the proportion of households represented at a town meeting is approximately 0.3913 to 0.4421.

uop;'u'