Question

find the margin of error for the given values of c, σ, and n.
c = 0.95, σ = 2.5, n = 64
click the icon to view a table of common critical values.
e = (round to three decimal places as needed.)

Explanation:

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

For a confidence level \( c = 0.95 \), the significance level \( \alpha=1 - c=1 - 0.95 = 0.05 \). Then \( \alpha/2=\frac{0.05}{2}=0.025 \).
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 \( \sigma = 2.5 \), \( n = 64 \), and \( z_{\alpha/2}=1.96 \).
Substitute the values into the formula: \( E=1.96\times\frac{2.5}{\sqrt{64}} \).
Since \( \sqrt{64}=8 \), then \( E = 1.96\times\frac{2.5}{8} \).
First calculate \( \frac{2.5}{8}=0.3125 \), then \( E=1.96\times0.3125 \).
\( E = 0.6125\approx0.613 \)

Answer:

\( 0.613 \)