Question

Venture-Capital Investments. In Exercise 8.69, you found a $95\%$confidence interval for the mean amount of all venture-capital investments in the fiber optics business sector to be from $\backslash (5.389$million to$\backslash )7.274$million. Obtain the margin of error by

a. taking half the length of the confidence interval.

b. using Formula $8.1$on page 325. (Recall that and that $\sigma =\$2.04$million.)

Short answer

(a) The Margin of Error is $0.9425$million.

(b) The Margin of Error is $0.9424$million.

Step-by-step solution

Given Information

It is given that upper confidence limit and lower confidence limit are $\$7.274\text{million,}\$5.389\text{million}$

$n=18$

$\sigma =\$2.04\text{million}$

Margin of Error by taking half of length of Confidence Interval

Length of Confidence Interval = Upper Confidence Limit - Lower Confidence Limit

$=\$7.274\text{million}-\$5.389\text{million}$

$=\$1.885\text{million}$

As Margin of Error is half the length of Confidence Interval

$=\frac{1}{2}\times \$1.885\text{million}$

$=\$0.9425\text{million}$

Hence, Margin of Error is$\$0.9425\text{million}$.

Determination of Margin of Error using Formula

For $95\%$confidence level $=100\times 0.95\%$

$\Rightarrow Z_{\frac{a}{2}}=Z_{0.025}$

$=1.96$

Formula for Margin of Error at $95\%$confidence level

$E=Z_{0.025}\times \left(\frac{\sigma }{\sqrt{n}}\right)$

$=\frac{1.96\times 2.04}{\sqrt{18}}$

$=0.9424$

Hence, Margin of Error is$0.9424$