Question

length of growing seasons the growing seasons for a random sample of 34 u.s. cities were recorded, yielding a sample mean of 188.4 days and the population standard deviation of 53.4 days. estimate the true population mean of the growing season with 95% confidence. round your answers to at least one decimal place.
□ < μ < □

Explanation:

Step1: Identify given values

Sample size $n=34$, sample mean $\bar{x}=188.4$, population SD $\sigma=53.4$, confidence level 95%

Step2: Find z-score for 95% confidence

$z_{\alpha/2}=1.96$ (standard 95% confidence z-score)

Step3: Calculate standard error

$SE=\frac{\sigma}{\sqrt{n}}=\frac{53.4}{\sqrt{34}}\approx\frac{53.4}{5.83095}\approx9.158$

Step4: Compute margin of error

$ME=z_{\alpha/2}\times SE=1.96\times9.158\approx17.95$

Step5: Find confidence interval

Lower bound: $\bar{x}-ME=188.4-17.95\approx170.45$; Upper bound: $\bar{x}+ME=188.4+17.95\approx206.35$ (rounded to one decimal)

Answer:

170.5 < μ < 206.4