THE ELECTION POLL REPORTED THAT A CANDIDATE HAD AN APPOROVAL RATIONG OF 30% WITH A MARGING OF ERROR E OF 4%. CONSTRUCT A CONFINDENCE INTERVAL FOR THE PROPORTIONS OF ADULTS WHO APPROVE OT THE CANDIDATE.
CI = .30 + or - .04
Therefore, interval is from .26 to .34 for the proportions.
Use the random-number table to simulate the outcomes of tossing a quarter 25 times. Assume that the quarter is balanced (i.e., fair). (Start at row 1, and let 0, 1, 2, 3, 4 indicate heads and 5, 6, 7, 8, 9 indicate tails. Enter H for heads and T for tails, and enter your answer as a comma-separated list.)
To construct a confidence interval for the proportion of adults who approve of the candidate, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
In this case, the sample proportion is given as 30% and the margin of error as 4%.
Step 1: Convert the percentage to a decimal
30% = 0.30
Step 2: Calculate the Margin of Error
Margin of Error = (z-score) * √[ (Sample Proportion * (1 - Sample Proportion)) / Sample Size]
To determine the z-score for a given confidence level, we need to consult a standard normal distribution table. Let's assume we are using a 95% confidence level, which corresponds to a z-score of approximately 1.96.
Margin of Error = 1.96 * √[ (0.30 * (1 - 0.30)) / Sample Size]
Step 3: Substitute the given values and solve for the Sample Size
0.04 = 1.96 * √[ (0.30 * (1 - 0.30)) / Sample Size]
Solving the equation for the Sample Size, we get:
Sample Size = (1.96^2 * 0.30 * (1 - 0.30)) / (0.04^2)
Sample Size ≈ 665.94
Since the sample size should be a whole number, we round it up to the nearest whole number. Therefore, the sample size needed is 666.
Step 4: Calculate the Confidence Interval
Using the given sample proportion of 30% and the calculated margin of error of 4%, we can now construct the confidence interval:
Confidence Interval = 0.30 ± 0.04
The confidence interval for the proportion of adults who approve of the candidate is approximately (0.26, 0.34) or 26% to 34% with a 95% confidence level.