Question

length of growing seasons the growing seasons for a random sample of 40 u.s. cities were recorded, yielding a sample mean of 192.5 days and the population standard deviation of 53.5 days. estimate the true population mean of the growing season with 99% confidence. round your answers to at least one decimal place.

<\mu <

Explanation:

Step1: Find the critical value $z_{\alpha/2}$

For a $99\%$ confidence interval, $\alpha = 1 - 0.99=0.01$, and $\alpha/2 = 0.005$.
From the standard normal distribution table, $z_{0.005} = 2.576$.

Step2: Calculate the margin of error $E$

The formula for the margin of error when the population standard deviation $\sigma$ is known is $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$.
Given $n = 40$, $\sigma=53.5$, $z_{\alpha/2}=2.576$.
$E = 2.576\times\frac{53.5}{\sqrt{40}}\approx2.576\times\frac{53.5}{6.3246}\approx2.576\times8.46\approx22.6$.

Step3: Calculate the confidence interval

The confidence interval for the population mean $\mu$ is given by $\bar{x}-E<\mu<\bar{x} + E$.
Given $\bar{x}=192.5$.
$\bar{x}-E=192.5 - 22.6=169.9$; $\bar{x}+E=192.5 + 22.6=215.1$.

Answer:

$169.9<\mu<215.1$