1. Using the Manip > Generate Random Data function on DataDesk,
generate 100 observations of a uniform random variable and call this variable
box1.
a) Make a histogram. Is it what you expect?
b) What's the average and SD of box1?
c) Now do the following:
i) draw a random sample with replacement of size
4. Find the average of this sample
ii) Repeat 50 times
iii) Make a histogram of the 50 averages
Hint: Here's a handout that describes how to do this.
d) Describe the shape of the histogram.
e) Based on your simulated data, what's the probability that an average
based on a sample size of 4 will be more than .15 units above the average
of box1?
f) Suppose that X was a normal random variable with mean equal to the
average of box1 and SD equal to (the SD of box1 divided by square root
of 4). (This is the SE of the average of four random samples.) Find
the probability that X will be more than .15 units above the average
of box1.
g) Are the answers to e and f close? Should they be?
h) Repeat (d) - (g), but now base each average on a sample (with
replacement) of size 16. Comment on the differences in yoru answers
with this larger sample size.
2. Using the Manip > Generate Random Data function, generate
100 observations from a Poisson distribution with lambda = 1. Call
this variable box2.
Repeat question 1 on this data. But now we add a part
i) Do the histograms of the 50 averages look similar or different for
the Poisson data and the Uniform data? Does it matter how many observations
used to make the average?
3. 1000 people are asked whether they believe Microsoft is a monopoly. Suppose that in the population it is known that 54% of the population believes this. Use the Central Limit Theorem to find the approximate probability that more than 70 people in this survey will believe Microsoft is a monopoly.
4. Suppose that the heights of women in the US follow a normal distribution
with mean 63.5 inches and SD 2.5 inches. If we take a random sample
of 10 people, find the probability that
a) 8 or more will be taller than 63.5 inches
b) 8 or more will be taller than 68 inches.
5. Suppose that womens' heights follows a N(63.5, 2.5) distribution.
We take a random sample of 16 women.
a) What is the probability that the average height of these 16 women
will be more than 63.5 inches?
b) What is the probability that the average height of these 16 women
will be within 1 inch of 63.5 inches?
c) How many women must we sample so that the probability that their
average height is within 1 inch of 63.5 inches is 95%?
d) Write the 5 steps of a simulation that will estimate the probability
asked for in part b. (This is the experimental probability.)
e) Carry out this simulation
6. Suppose that we treat the students in this class as a "population",
and we wish to estimate the mean height of this population. Assume
that the heights follow a normal distribution.
a) Take a random sample of size 9 with replacement from the class
data. Use this to estimate the mean height.
b) Assuming the population is normally distributed, write the
distribution for an average based on a random sample of size 9.
c) Estimate the standard deviation of the population.
d) Find a 95% confidence interval for the mean.
e) Take 100 random samples of size 9 and find their averages.
Make a histogram. Do the averages look like the distribution you
gave for part b?
f) Find the standard deviation of the averages. Use
this to re-compute the 95% confidence interval you found in (d).
Does it make a big difference?
g) Open up the "box": What is the population mean and standard
deviation? Did your confidence interval cover it?
STM: p. 448: 5,6,7
p. 460: 6,7
p.462: 1, 3