Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): $60.2,47.0,235.1,490.0$, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield $\bar { x } = 180.975$ and $s = 143.042$. Calculate a $95 \%$ confidence interval for the mean endowment of all private colleges in the United States.
A) $180.975 \pm 116.621$
B) $180.975 \pm 119.605$
C) $180.975 \pm 127.863$
D) $180.975 \pm 124.673$

Question

Quizplus · Accepted Answer

To calculate a 95% confidence interval for the mean, we use the formula: $\bar{x} \pm t_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\frac{\alpha}{2}}$ is the t-score for a 95% confidence level with $n-1$ degrees of freedom. Given $\bar{x} = 180.975$, $s = 143.042$, and $n = 8$, we find $t_{\frac{\alpha}{2}}$ for 7 degrees of freedom (since $n-1 = 8-1 = 7$) at a 95% confidence level. The t-score for 7 degrees of freedom at a 95% confidence level is approximately 2.365. Plugging these values into the formula gives us the margin of error: $2.365 \times \frac{143.042}{\sqrt{8}} \approx 119.605$. Therefore, the 95% confidence interval is $180.975 \pm 119.605$.

Private Colleges and Universities Rely on Money Contributed by Individuals 60.2,47.0,235.1,490.060.2,47.0,235.1,490.060.2,47.0,235.1,490.0

Private Colleges and Universities Rely on Money Contributed by Individuals $60.2,47.0,235.1,490.0$