A US travel data center survey of 1500 adults found that 42% of the respondents stated that they favor historical sites as vacations find the 95% confidence interval of the troop portion of all adults who favor historical sites as vacation
To find the 95% confidence interval for the proportion of all adults who favor historical sites as vacation, we can use the following formula:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
Where:
- Sample Proportion is the proportion of respondents in the survey who favor historical sites (42% or 0.42 in decimal form).
- Z is the Z-score associated with the desired level of confidence. For a 95% confidence level, the corresponding Z-score is approximately 1.96.
- Standard Error is the measure of the variability or spread in the sample proportion and is calculated as the square root of [(Sample Proportion * (1 - Sample Proportion)) / Sample Size].
Given that the survey had a sample size of 1500 adults and the sample proportion is 0.42, we can calculate the confidence interval.
Step 1: Calculate the Standard Error
Standard Error = sqrt[(0.42 * (1 - 0.42)) / 1500]
Step 2: Calculate the Margin of Error (Z * Standard Error)
Margin of Error = 1.96 * Standard Error
Step 3: Calculate the Confidence Interval
Confidence Interval = Sample Proportion ± Margin of Error
Now, let's calculate the confidence interval:
Step 1: Calculate the Standard Error
Standard Error = sqrt[(0.42 * (1 - 0.42)) / 1500]
Standard Error ≈ 0.013
Step 2: Calculate the Margin of Error (Z * Standard Error)
Margin of Error = 1.96 * 0.013
Margin of Error ≈ 0.025
Step 3: Calculate the Confidence Interval
Confidence Interval = 0.42 ± 0.025
Therefore, the 95% confidence interval for the proportion of all adults who favor historical sites as vacations is approximately 0.395 to 0.445.