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 Upper X overbarequals180.975 and Sequals143.042. 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.

So on this case the 95% confidence interval would be given by (61.370;300.580)  

Step-by-step explanation:

Previous concepts

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

Solution to the problem

represent the sample mean  

population mean (variable of interest)  

represent the sample standard deviation  

n=8 represent the sample size  

The confidence interval on this case is given by:

  (1)

We can find the degrees of freedom and we got:

The next step would be find the value of , and  

Using the t table with df =7, excel or a calculator we see that:  

Since we have all the values we can replace:

So on this case the 95% confidence interval would be given by (61.370;300.580)  

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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