Question

length of growing seasons the growing seasons for a random sample of 37 u.s. cities were recorded, yielding a sample mean of 188.4 days and the population standard deviation of 55.4 days. estimate the true population mean of the growing season with 95% confidence. round your answers to at least one decimal place.

Explanation:

Step1: Find the z - value

For a 95% confidence interval, the significance level \(\alpha=1 - 0.95=0.05\). Then \(\alpha/2=0.025\). Looking up in the standard normal distribution table, \(z_{\alpha/2}=z_{0.025}=1.96\).

Step2: Calculate the margin of error \(E\)

The formula for the margin of error when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\times\frac{\sigma}{\sqrt{n}}\). Given \(n = 37\), \(\sigma=55.4\), and \(z_{\alpha/2}=1.96\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=188.4\)

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Answer:

\(170.6<\mu<206.3\)