Question

The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for

candidates and political committees each election cycle. A political action committee (PAC) is a committee formed to raise

money for candidates and campaigns. A Leadership PAC is a PAC formed by a federal politician (senator or representative)

to raise money to help other candidates’ campaigns.

The FEC has reported financial information for $556$Leadership PACs that operating during the $2011–2012$election cycle.

The following table shows the total receipts during this cycle for a random selection of $30$Leadership PACs.

̄$\overline{x}$$=\backslash (251,854.23$

$s=\backslash )521,130.41$

Use this sample data to construct a $96\%$confidence interval for the mean amount of money raised by all Leadership PACs

during the $2011–2012$election cycle. Use the Student's t-distribution.

Short answer

We estimate that the mean amount of money raised by all Leadership PACs during the $2011-2012$election cycle was between $\$47,292.57and\$456,415.89$with a$96\%$ confidence level.

Step-by-step solution

Given Information 

Create a $90\%$ confidence interval for the mean amount of money raised by all Leadership PACs during the $2011-2012$ election cycle using this sample data. The Student's-$t$ distribution should be used.

Explanation 

Now we start calculation:

$$\begin{gathered} \overline{x}=\$251,854.23, \\ s=\$521,130.41 \end{gathered}$$

The sample has a total of$30$measurements.

$n=30,\text{and}df=30-1=29$

$CL=0.96$

$\alpha =1-CL$

$=1-0.96$

$=0.04\frac{a}{2}$

$=0.02t_{\frac{a}{2}}$

$=t_{0.02}$

$=2.150$

$EBM=t_{\frac{a}{2}}\left(\frac{s}{\sqrt{n}}\right)$

$EBM=2.150\left(\frac{521,130.41}{\sqrt{30}}\right)$

$EBM=\$204,561.66$

$\overline{x}-EBM=\$251,854.23-\$204,561.66$

$\overline{x}-EBM=\$47,292.57$

$\overline{x}+EBM=\$251,854.23+\$204,561.66$

$\overline{x}+EBM=\$456,415.89$

We estimate that the average amount of money raised by all Leadership PACs during the $2011-2012$election season was between $\$200,000and\$300,000,$with a $96\%$confidence level$\$47,292.57and\$456,415.89$.

Final result 

We estimate that the mean amount of money raised by all Leadership PACs during the $2011-2012$election cycle was between $\$47,292.57and\$456,415.89$with a$96\%$ confidence level.