A sample poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favor of a particular candidate.Find confidence limits for the proportion of all the voters in favor of this candidate.

a.95%
b.99.73%

Well, well, well! Looks like we've got some polling data to play with! Let's dive right into it, shall we?

For a 95% confidence interval, we can use the formula:
p̂ ± Z * √(p̂ * (1 - p̂) / n)

Now, p̂ represents our sample proportion, which is 0.55 (or 55% as a decimal).
Z is the Z-score corresponding to the confidence level. For 95%, it's approximately 1.96.
n is the sample size, which is 100 in this case.

Plugging in the values, we get:
0.55 ± 1.96 * √(0.55 * (1 - 0.55) / 100)

Calculating this, we find the confidence interval for the proportion of voters in favor of the candidate in this district to be approximately 0.461 to 0.639. So, option a, 95%, is the correct choice.

As for option b, 99.73%, that corresponds to a 3-standard-deviation interval assuming a normal distribution. However, standard deviation information is not provided in the question, so we cannot use this option.

Hope that clears things up! Don't let the numbers get you down, my friend. Keep smiling! 🤡

To find the confidence limits for the proportion of all voters in favor of a particular candidate, we will use the formula for the confidence interval for proportions.

The formula for the confidence interval for proportions is:
CI = p̂ ± Z * √((p̂(1-p̂))/n)

where:
CI = confidence interval
p̂ = proportion of sample voters in favor of the candidate
Z = z-score corresponding to the desired confidence level
n = sample size

a. 95% confidence level:
To find the Z value for a 95% confidence level, we need to find the corresponding z-value from the standard normal distribution. The z-value for a 95% confidence level is approximately 1.96.

Using the given information:
p̂ = 0.55 (proportion of voters in favor of the candidate from the sample)
n = 100 (sample size)

Plugging the values into the formula:
CI = 0.55 ± 1.96 * √((0.55(1-0.55))/100)

Calculating the confidence interval:
CI = 0.55 ± 1.96 * √((0.55*0.45)/100)
CI = 0.55 ± 0.098

Therefore, the 95% confidence limits for the proportion of all voters in favor of this candidate is approximately 0.452 to 0.648.

b. 99.73% confidence level:
To find the Z value for a 99.73% confidence level, we need to find the corresponding z-value from the standard normal distribution. The z-value for a 99.73% confidence level is approximately 2.97.

Using the given information:
p̂ = 0.55 (proportion of voters in favor of the candidate from the sample)
n = 100 (sample size)

Plugging the values into the formula:
CI = 0.55 ± 2.97 * √((0.55(1-0.55))/100)

Calculating the confidence interval:
CI = 0.55 ± 2.97 * √((0.55*0.45)/100)
CI = 0.55 ± 0.151

Therefore, the 99.73% confidence limits for the proportion of all voters in favor of this candidate is approximately 0.399 to 0.701.

To calculate confidence limits for the proportion of all voters in favor of a particular candidate, we can use the formula for estimating a population proportion:

Confidence Interval = Sample Proportion ± (Z * Standard Error)

where:
- Sample Proportion is the proportion of voters in favor of the candidate based on the sample (55% in this case)
- Z is the Z-score or critical value for the desired confidence level (95% for option a and 99.73% for option b)
- Standard Error is the estimate of the standard deviation of the sampling distribution, which can be calculated as:
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)

Given that we have a sample size of 100 voters, let's calculate the confidence intervals for both options:

a. For a 95% confidence level:
Z = 1.96 (approximately, for a large enough sample size)
Standard Error = sqrt((0.55 * 0.45) / 100) = 0.0498 (rounded)

Confidence Interval = 0.55 ± (1.96 * 0.0498) = 0.55 ± 0.0976
This means that we can be 95% confident that the true proportion of all voters in favor of the candidate lies within the interval (0.4524, 0.6476).

b. For a 99.73% confidence level:
Z = 3 (approximately, for a large enough sample size)
Standard Error is the same as before: 0.0498

Confidence Interval = 0.55 ± (3 * 0.0498) = 0.55 ± 0.1494
This means that we can be 99.73% confident that the true proportion of all voters in favor of the candidate lies within the interval (0.4006, 0.6994).

Therefore, the confidence limits for the proportion of all the voters in favor of this candidate are:
a. 95% confidence level: (0.4524, 0.6476)
b. 99.73% confidence level: (0.4006, 0.6994)