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### Problems: The Central Limit Theorem Part 1

1. Suppose that the average age of books in the library is 37 years with a standard deviation of 10 years. Assume further that the distribution is normal.

a) If one selects a book at random, what is the probability that it will be less than 32 years old?
b) Suppose that a random sample of 25 books is selected and the mean age of the sample computed. What is the probability that this mean will be less than 32?
c) Between what two ages would we find the middle 95% of book ages when we are looking at individual books from this population?
d) Between what two ages would we find the middle 95% of all sample means when the sample size is 100?

2. Consider the following probability distribution:

 x P(x) 0 .2 1 .2 2 .2 3 .2 4 .2

The mean (which is the same thing as expected value) of this distribution is 2 and that the standard deviation is the square root of 2 (or 1.4121...)

a) Show how one discovers that the mean of this distribution is in fact 2.
b) If we take a random sample of 32 from this distribution and find the sample mean, what is the probability that the result will be between 1.5 and 2.5?
c) If we take random sample of one from this distribution, what is the probability that the value we will get will be between 1.5 and 2.5? (Hint: if you see it, it is very easy--no computation is required.)
d) (not so easy--you do not use the normal distribution--you work this like we worked dice problems--draw the set of all equally probably outcomes.) If we take a random sample of two from this distribution, what is the probability that the sample mean will be equal to or greater than 1.5 and less than or equal to 2.5?

3. A Christmas tree farm planted 100 acres of trees five years ago. These trees now have an average height of 80 inches with a standard deviation of 5 inches.

a) Assuming the distribution is normal, suppose one tree is picked at random. What is the probability that it will be less than 75 inches?
b) Suppose a random sample of 25 trees is selected and the average height computed. What is the probability that it will be less than 75 inches?
c) What is the probability that one tree selected at random will be within 1 inch of the true mean?
d) What is the probability that the mean of a random sample of 100 trees will be within 1 inch of the true mean?

4. Early in the last century, sports fishermen introduced non-native species of trout into most of the streams in the West and these non-native species eliminated the native species. In recent years agencies of the government have tried to undo what was done by extirpating the introduced species and re-introducing the native species into some of these streams.

Suppose that a year after reintroduction, the Bonneville cutthroat trout have an average length of 4 inches with a standard deviation of .4 inches.

a) Assuming that length is normally distributed, if we selected at random a trout from this population, what is the probability that it would be longer than 4.1 inches?
b) If we randomly select 16 trout from this population, what is the probability that the sample mean will be larger than 4.1 inches?
c) Suppose that the people doing the sampling do not know what the true mean is. However, the people re-introducing these fish into the stream had a goal that the average length of the fish would be 4 inches after a year. A sample mean of 100 trout is 3.9 inches with a standard deviation of .4 inches. Should they worry that the goal has not been met? Explain.

5. A biologist is coring bristlecone pine trees on Wheeler Peak in Eastern Nevada to determine their age. Suppose that the average age of the trees in this area is 2500 years with a standard deviation of 500 years.

a) Assuming that the population is normally distributed, how old would a tree have to be in order to be in the oldest 5% of these trees?
b) Suppose the biologist could take many samples of size 25 from this population and compute the sample means. Above what age would he expect to find the largest 5% of sample means?

6. Suppose that the average age of people who die is 86 and the standard deviation is 10 years.

a) If we take a random sample of 25 deaths, what is the probability that our sample mean age will be between 85 and 87?
b) If we take a random sample of 100 deaths, what is the probability that our sample mean age will be between 85 and 87?

7. A box with a million tickets has an average of 60 with a standard deviation of 5. If a sample of 400 draws is made from this box, we expect an average of ______________ give or take ____________or so. The probability that the average would be less than 59 is ________________________.

8. Suppose that you are fairly sure that you are dealing with a random variable that is normally distributed with a mean of 5 and a standard deviation of 1. Is it likely that such a distribution would yield a random sample of these numbers: {4, 5, 5, 5, 6, 6, 7, 7, 8, 8}? Explain carefully how you get your answer.

9. The expected value of the random digits (zero to nine) is 4.5, their standard deviation is 2.67, and they are distributed uniformly.

a) Suppose a sample of 25 random digits it taken. What do we know about the properties of the sample mean, x-bar?
b) Use these properties to calculate the approximate percentage of the time that the sample mean will be between 4 and 5.
c) Repeat b assuming that the sample size is 100.

10. Suppose that you had a million dollars to invest but you had to invest it in one of two options. In option A you have a 40% chance of doubling you money in a year, a 40% chance of just getting back your million dollars, and a 20% chance of losing it all. With option B you are sure that in one year you will receive \$1.1 million.

a) What is the expected value of each option?