Question

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an​ endowment, and the college spends only the interest earned by the fund. A recent survey of 8 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 180.975 and s143.042. Calculate a​ 95% confidence interval for the mean endowment of all the private colleges in the United States assuming a normal distribution for the endowments. Round to three decimal places.

Answer 1

Answer:

The 95% confidence interval   $81.852 < \mu < 280.098$

Step-by-step explanation:

From the question we are told that

   The  sample mean is  $\= x = 180.975$

   The standard deviation is  $\sigma = 143.042$

   The sample size is n = 8

Given that the confidence level is  95%  then the level of significance is  

    $\alpha =( 100- 95) \%$

=>  $\alpha =0.05$    

Generally from the normal distribution table the critical value  of  $\frac{\alpha }{2}$ is  

   $Z_{\frac{\alpha }{2} } = 1.96$

Generally the margin of error is mathematically represented as  

      $E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }$

=>  $E = 1.96 * \frac{ 143.042 }{\sqrt{8} }$

=>  $E = 99.123$

Generally 95% confidence interval is mathematically represented as  

      $\= x -E < \mu < \=x +E$

=>    $180.975 -99.123 < p < 180.975 +99.123$

=>    $81.852 < \mu < 280.098$