Final answer:
The confidence interval to estimate the true population mean of the growing season is 178.43 to 212.97 days with 95% confidence.
Step-by-step explanation:
The question involves estimating the true population mean of the growing season with 95% confidence based on a sample mean and the population standard deviation using the sample data of 40 U.S. cities. To achieve this, we utilize the formula for a confidence interval when the population standard deviation is known, which is the sample mean ± (Z-score * (population standard deviation / sqrt(sample size))). With a 95% confidence interval, the Z-score associated is typically 1.96.
Substituting the given values:
- Sample mean (μ) = 195.7 days
- Population standard deviation (σ) = 55.7 days
- Sample size (n) = 40
- Z-score for 95% confidence = 1.96
The confidence interval calculation is as follows:
195.7 ± (1.96 * (55.7 / sqrt(40)))
Firstly, calculate the standard error (SE): SE = 55.7 / sqrt(40) = 8.81 (rounded off to two decimal places)
Then, calculate the margin of error (MOE): MOE = 1.96 * SE = 1.96 * 8.81 = 17.27 (rounded off to two decimal places)
This results in a confidence interval of:
195.7 - 17.27 to 195.7 + 17.27
178.43 to 212.97 days (rounded to at least one decimal place)