Q 8.87 - Elementary Statistics

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.)

Step-by-Step Solution

Answer

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

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

Step 1: Given Information

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

$n=18$

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

Step 2: 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 $\$0.9425\text{ million }$.

Step 3: 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 $0.9424$