Question

The Township Board of Meridian Township wants to know how much public support there is for raising property taxes to fix and maintain the roads in the township. They randomly surveyed Township residents and calculated a 99% confidence interval for the actual proportion of Meridian Township residents who are in favor of the tax increase to be (0.55, 0.62) 1. What is the sample proportion? Give the exact answer as a decimal 2. Which of the following statements must be true? A A 95% confidence interval calculated using the same survey data will include fewer plausible values for the population proportion B. If the sample size had instead been 870 residents, then the 99% confidence interval would be half as wide as the one stated above OC. If a different sample of 435 residents were to be selected, then there is a 99% chance that the new sample proportion will be between 0.55 and 0.62 3. What is the margin of error for this confidence interval? Give the exact answer as a decimal. 4. What is the standard error for this confidence interval? Round your answer to 4 decimal places, 5. What is the 90% confidence interval using the same survey data?

Answer 1

The sample proportion of Meridian Township residents who are in favor of the tax increase is 0.585. The margin of error for the given 99% confidence interval is 0.035. A 90% confidence interval using the same data cannot be calculated from the information given without knowing the standard error and the sample size.

Step-by-step explanation:

The sample proportion is the midpoint of the confidence interval. In this case, it is the average of the lower and upper limits (0.55 and 0.62), so the sample proportion is (0.55 + 0.62)/2 = 0.585.

Regarding the statements, B is incorrect because the width of a confidence interval depends on both the sample size and the standard error, not just the sample size. A is incorrect because a 95% confidence interval would be narrower, not including fewer plausible values, and C is incorrect because there is not a 99% chance that a new sample's proportion will fall between 0.55 and 0.62; the confidence interval only applies to the population proportion.

The margin of error for this confidence interval can be calculated as half the width of the interval: (0.62 - 0.55)/2 = 0.035.

The standard error cannot be calculated directly from the information provided, as it requires knowing the sample size. Without the sample size or the formula used to calculate the confidence interval, we cannot determine the standard error.

To calculate a 90% confidence interval using the same survey data, we would need to know the standard error and the appropriate z-score for a 90% confidence level. Without the standard error, we cannot calculate the 90% confidence interval.