The Central Limit Theorem

Instructions:

                     1)   Show your work.  For calculator problems, record the command(s) you used.

                     2)   Do your calculations using formulas unless otherwise indicated. 

            1.  Fill in the blanks:

                 The Central Limit Theorem applies when the original population is _________________ distributed,

                  or n ______________, or both.

            2.  Sketch the graph of a uniform distribution with values ranging from 5.0 to 20.0, and label the height.

            3.  A population consists of a $4 pen, a $16 pen and a $10 pen.  Create a list of all samples of size 2

                taken with replacement, and use it to find the sampling distribution of the sample means. 

            4.  The length of time a particular surgery takes is normally distributed with a mean of  53.5 minutes and

                 a standard deviation of 7.9 minutes.)

            a.  If one surgery is randomly selected, find the probability that it takes more than 1 hour.  Use a chart.

            b.  If 36 surgeries are randomly selected, find the probability that their mean length is more than 1 hour. 

                 Use your calculator.

c.  Find , the length of time it takes for 90% of surgeries to be completed.  You may use your

                calculator.

            6.  A charity holds an annual raffle where the probability of winning is .003.

            a.  If you enter the raffle every year for the next 20 years, find the probability that you will win exactly

                once.  You may use your calculator.

            b.  If you enter the raffle every year for the next 20 years, find the probability that you will win at least

                 once.

            c.  Use the range rule of thumb to estimate the usual number of wins for people who enter the raffle each

                 year, for 20 years. 

            d.  In the raffle, you could lose $50 or win $10,000.  Find the expected value of your winnings, and

                describe the meaning of that value.

            7.  A basket contains 6 apples, 4 oranges and 10 bananas.

a.  If one piece of fruit is randomly selected from the basket, find the probability that it is an apple.

            b.  If two pieces of fruit are randomly selected from the basket without replacement, find the probability

                 that the second one is a banana given that the first one was an orange.

            c.  If three pieces of fruit are randomly selected from the basket without replacement, find the

                 probability that they are all apples.

            d.  If three pieces of fruit are randomly selected from the basket with replacement, find the probability

                 that they are all apples.

            8.  The type of businesses that opened in the last year in a community are described in the chart.  If one

                 business is randomly selected, find the probability that…

            a.  it is a restaurant or it failed.

            b.  it is a restaurant and it failed.

            c.  it is a restaurant given that it failed.