Download Suppose we have a box that contains tickets with values that range

Lab 8
Suppose we have a box that contains tickets with values that range from 0 to 1. There are
many, many tickets in this box and each ticket has its own unique value (no 2 tickets are
identical). Each ticket has the same chance of being drawn from the box as any other
ticket. This description describes the uniform distribution.
1. If you could compute the average of all of the tickets in the box, what would the
average be?
2. Is the average of all of the tickets in the box a population parameter (population
average) or a sample statistic (sample average)?
Let’s create the probability histogram for the tickets in the box.
3. What is the name of that probability histogram?
Type
Net from http://www.ats.ucla.edu/stat/stata/ado
Net install clt
Clt, samples(50000) n(1) dist(u)
[draw 1 ticket with replacement from the
box of tickets which has a uniform
distribution, 50000 times]
4. Describe the shape of the distribution. Does it look like a normal curve? Is the mean
of that probability histogram the same value as the population average? If not exactly,
why not? Jot down the standard deviation of the distribution.
Let’s imagine that we are going to draw some tickets at random with replacement from
this box and compute the average value of those tickets.
5. Is the average of the tickets in the sample a population parameter or a sample statistic?
Now let’s imagine that we are going to repeat the process of drawing some tickets at
random with replacement from this box an infinite number of times, compute the sample
average each time, and create a probability histogram of all of those sample means.
6. What is the name of that probability histogram?
We can approximate a theoretical sampling distribution of sample means from the box
described above by taking many SRSs, like 50,000. And we can choose to repeatedly
draw a few as 2 tickets at random with replacement, or we can choose to repeatedly draw
many more tickets at random with replacement. Unfortunately, we are limited to
repeating the process 50,000 times rather than an infinite number of times.
Type
clt, samples(50000) n(2) dist(u) normal
[repeat the process 50,000 times, draw 2
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
7. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
Type
clt, samples(50000) n(20) dist(u) normal
[repeat the process 50,000 times, draw 20
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
8. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
Type
clt, samples(50000) n(100) dist(u) normal
[repeat the process 50,000 times, draw 100
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
9. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
10. At what sample size did the sampling distribution of sample means look like the
normal curve?
Now let’s repeat the whole process using a different population distribution. This time
imagine that if you created a probability histogram of the tickets in the box, that the
distribution would look like a normal curve with mean = 0 and standard deviation = 1.
11. If you could compute the average of all of the tickets in the box, what would the
average be?
Let’s create the probability histogram for the tickets in the box.
12. What is the name of that probability histogram?
Type
Clt, samples(50000) n(1) dist(n) normal
[draw 1 ticket with replacement from
the box of tickets which has a normal
distribution, 50000 times]
13. Describe the shape of the distribution. Does it look like a normal curve? Is the mean
of that probability histogram the same value as the population average? If not exactly,
why not? Jot down the standard deviation.
Now let’s imagine that we are going to repeat the process of drawing some tickets at
random with replacement from this box an infinite number of times, compute the sample
average each time, and create a probability histogram of all of those sample means.
14. What is the name of that probability histogram?
Type
clt, samples(50000) n(2) dist(n) normal
[repeat the process 50,000 times, draw 2
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
15. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
Type
clt, samples(50000) n(20) dist(n) normal
[repeat the process 50,000 times, draw 20
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
16. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
Type
clt, samples(50000) n(100) dist(n) normal
[repeat the process 50,000 times, draw 100
tickets at random with replacement, compute
the sample average each time, plot the
histogram of the 50,000 sample averages]
17. Describe the shape of the distribution. Does it look like a normal curve? What is the
mean of this distribution? Is it very close to the population mean? If it’s off a bit, why?
Compute the standard deviation of this distribution using the information about the
standard deviation from the population distribution. Do you get the same standard
deviation as the computer program got?
18. At what sample size did the sampling distribution of sample means look like the
normal curve?
19. Please describe:
a) how this lab illustrates the central limit theorem
b) the conclusion that you draw about the value of the population average and
the average of the sampling distribution of sample means.
What to turn in: all the numbered sections.