Question

a random sample of 56 fluorescent light bulbs has a mean life of 645 hours and assume that the population standard deviation is 31 hours. construct a 95% confidence interval for the population mean.
a. (539.6,551.2)
b. (712.0,768.0)
c. (112.0,118.9)
d. (636.9,653.1)

Explanation:

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

For a 95% confidence interval, \( \alpha=1 - 0.95=0.05 \), and \( \alpha/2=0.025 \).
From 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 \( E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \).
Given \( n = 56 \), \( \sigma=31 \), and \( z_{\alpha/2}=1.96 \).
\( E=1.96\times\frac{31}{\sqrt{56}}\)
\(=\frac{1.96\times31}{7.4833}\)
\(=\frac{60.76}{7.4833}\approx8.1\)

Step3: Construct the confidence interval

The confidence interval for the population mean \( \mu \) is \( \bar{x}-E<\mu <\bar{x}+E \).
Given \( \bar{x} = 645 \), \( \bar{x}-E=645 - 8.1=636.9 \), \( \bar{x}+E=645+8.1 = 653.1 \)

Answer:

D. \( (636.9,653.1) \)