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%
(a) In a poll of 10,000 voters selected at random from all voters in a certain district, it is found that 5,180 voters are in favor of particular candidate. Test the null hypothesis that the proportion of all the voters in the district, who favor the candidate is equal to or less than 50% against the alternative that it is greater than 50%. Use a 0.05 level of significance.
To find the confidence limits for the proportion of all voters in favor of a particular candidate, we can use the formula for confidence intervals for proportions.
a. 95% Confidence Interval:
Step 1: Calculate the standard error (SE):
SE = sqrt((p * (1 - p)) / n)
where p is the sample proportion (55% or 0.55) and n is the sample size (100).
SE = sqrt((0.55 * (1 - 0.55)) / 100) = 0.0497
Step 2: Find the margin of error (ME):
ME = z * SE
where z is the z-score corresponding to the desired confidence level. For a 95% confidence level, the z-score is approximately 1.96.
ME = 1.96 * 0.0497 = 0.0973
Step 3: Calculate the lower and upper limits of the confidence interval:
lower limit = p - ME
upper limit = p + ME
lower limit = 0.55 - 0.0973 = 0.4527
upper limit = 0.55 + 0.0973 = 0.6473
Therefore, the 95% confidence interval for the proportion of all voters in favor of the candidate is approximately 0.4527 to 0.6473.
b. 99.73% Confidence Interval:
To calculate the 99.73% confidence interval, we need to find the z-score corresponding to this confidence level. The z-score is approximately 3.
ME = z * SE
ME = 3 * 0.0497 = 0.1491
lower limit = p - ME
lower limit = 0.55 - 0.1491 = 0.4009
upper limit = p + ME
upper limit = 0.55 + 0.1491 = 0.6991
Therefore, the 99.73% confidence interval for the proportion of all voters in favor of the candidate is approximately 0.4009 to 0.6991.
To find confidence limits for the proportion of all voters in favor of a particular candidate, we can use the formula for the confidence interval:
Confidence interval = sample proportion ± (critical value * standard error)
In this case, the sample proportion is 55%, and we need to find the confidence limits for a 95% confidence interval and a 99.73% confidence interval.
a. For a 95% confidence interval:
Step 1: Find the critical value - the critical value for a 95% confidence interval is approximately 1.96. This can be obtained from a standard normal distribution table or by using statistical software.
Step 2: Calculate the standard error - the standard error is given by the formula:
Standard error = √((sample proportion * (1 - sample proportion)) / sample size)
In this case, the sample proportion is 0.55, and the sample size is 100. Plugging these values into the formula, we get:
Standard error = √((0.55 * (1 - 0.55)) / 100) ≈ 0.048
Step 3: Calculate the confidence interval:
Confidence interval = 0.55 ± (1.96 * 0.048) = 0.55 ± 0.094
The lower confidence limit is 0.55 - 0.094 = 0.456, and the upper confidence limit is 0.55 + 0.094 = 0.644.
Therefore, the confidence limits for the proportion of all voters in favor of the candidate, with 95% confidence, are between 45.6% and 64.4%.
b. For a 99.73% confidence interval:
Step 1: Find the critical value - the critical value for a 99.73% confidence interval is approximately 3.08.
Step 2: Calculate the standard error - using the same formula as in part a, we get:
Standard error = √((0.55 * (1 - 0.55)) / 100) ≈ 0.048
Step 3: Calculate the confidence interval:
Confidence interval = 0.55 ± (3.08 * 0.048) = 0.55 ± 0.148
The lower confidence limit is 0.55 - 0.148 = 0.402, and the upper confidence limit is 0.55 + 0.148 = 0.698.
Therefore, the confidence limits for the proportion of all voters in favor of the candidate, with 99.73% confidence, are between 40.2% and 69.8%.