Question

How much money do people spend on graduation gifts? In 2007, a federation surveyed 2415 consumers who reported that they bought one or more graduation gifts that year. The sample was selected in a way designed to produce a sample representative of adult Americans who purchased graduation gifts in 2007. For this sample, the mean amount spent per gift was $58.15. Suppose that the sample standard deviation was $20. Construct a 98% confidence interval for the mean amount of money spent per graduation gift in 2007. (Round your answers to three decimal places.)
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Interpret the interval.

We are 98% confident that the mean amount of graduation money spent was within this interval.

We are 98% confident that the mean amount of money spent per graduation gift in 2007 was within this interval.    

We are confident that the mean amount of money spent per graduation gift in 2007 was within this interval 98% of the time.

We are confident that 98% of the amount of money spent per graduation gift in 2007 was within this interval.

Answer 1

Let us assume that our sample( x₁, x₂, ....., x_n) is drawn from normal population.

So , x_i ~ and mean of all sample values will be and  

~

Thus 1- % confidence interval of    is given by

Here 58.15

sample standard deviation is 20

When is unknown and , we take sample standard deviation as the estimator of  

That is

n=2415

is our confidence interval

Using this values we have ,

Lower confidence limit = = 58.131

Upper confidence limit = = 58.169

Hence confidence interval is (58.131, 58.169)

This would be the interpretation:

We are confident that the mean amount of money spent per graduation gift in 2007 was within this interval 98% of the time.

The confidence interval you compute depends on the data you have collected. If you repeat the experiment, your confidence interval would almost certainly be different.If multiple samples were drawn from the same population and a 98% Confidence intervals calculated for each sample, we would expect the population mean to be found within 98% of these Confidence intervals.