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Sampling Distributions for Means
Worksheet 2
Open the
interactive version of the normal tables. (Opens a separate browser window.)
Obtain some pictures of normal curves. Print them out and use them to help
solve problems.
Note: The typical symbol for the sample mean, an X
with an overbar, doesn't integrate well into web
documents. Therefore, I've merely refered to it as either
"X-bar" (what it's often called, anyway) or
"the sample mean" (what it is).
1. A jar of Jiffy Peanut Butter (labeled as a 32 oz.
jar) is selected randomly off the end of the assembly
line. The weight of the jar is measured. If X measures
this weight then X has the N(32.3, 0.40) distribution
(the filling machine is calibrated this way). That is,
the mean fill is 32.3 oz., the standard deviation of the
fills is 0.40 oz..
| a) |
Identify the response variable
being measured. Is it categorical or
quantitative? |
| b) |
Identify the population and
parameters. |
| c) |
What is the average fill of all
Jiffy jars? Is this value a statistic? |
| d) |
What's the probability a
randomly selected jar is filled with less than
32.0 fl oz? Such a can would be
"under-volume." |
| e) |
What proportion of all Jiffy
jars are under-volume? |
| f) |
Consider a simple random sample
of 6 jars. Let X-bar be the sample mean weight
for these 6 jars. Is X-bar a parameter or a
statistic? |
| g) |
What are the mean and standard
deviation of the (sampling) distribution for the
sample mean? |
| h) |
What is the shape of the sample
means's distribution? |
| i) |
Find the probability the sample
mean fill of the six jars is less than 32.0, P[
X-bar < 32.0]. |
| j) |
Now take a simple random sample
of 24 jars. (The sample size has been multiplied
by 4.) What are the mean and standard deviation
of the distribution for X-bar? How has the
standard deviation of changed with the
quadrupling of the sample size? |
| k) |
Find the probability the sample
mean fill of the 24 jars is less than 32.0. |
2. The weights, in pounds, of Portuguese Water Dogs
has the N(45, 3) distribution.
| a) |
If one dog is selected at random
and X is its weight, find the probability X is
within 1 pound of the mean weight of 45, P[44
< X < 46]. |
| b) |
Draw a simple random sample of
40 dogs. Find the probability the sample mean
weight is within 1 pound of the mean weight of
45. |
| c) |
Draw a simple random sample of
80 dogs. Find the probability the sample mean
weight is within 1 pound of the mean weight. |
3. The number of murders in Oswego County in a year is
a random variable. Suppose the mean of the distribution
of yearly murders is 1.8 and the standard deviation is
1.5.
| a) |
Let X represent the number of
murders in some (past, present or future) year.
Does X have a normal distribution? If not, what
distinguishes the distribution of X from a normal
distribution? (Skew? Discreteness?) |
| b) |
Suppose we are interested in
predicting the number of murders in future years
(jail cells for these dangerous criminals must be
built in advance). Take for instance the next
five years. Let X-bar be the average yearly
figure for the next five years (assume the values
for the individual years are independent). What
are the mean and standard deviation of X-bar?
(The sample size and the hypothesized shape of
the distribution for X lead me to conclude that
the Central Limit Theorem should not be applied
in order to approximate probabilities involving
for n = 5.) |
| c) |
Suppose instead n = 25. Using
the Central Limit Theorem, compute the
approximate probability that the sample mean
number of murders for the 25 years is above 2.5.
(To solve the problem first find the mean and
standard deviation of the sampling distribution
for when n = 25.) |
| d) |
If the sample mean number of
murders over the next 25 years exceeds 2.5, what
can you say about the total number of murders
over the same period? |
4. Stock market analysts measure the risk of a given
company's stock using an quantity called the "b index." (How this value is
computed is not important-it just represents one way
stock analysts measure a stock's risk.) A company with b > 1 has stock with above
average investment risk; b
< 1 means it has below average risk (its a relatively
safe investment). The distribution of this variable (for
companies listed on the New York Stock Exchange) has mean
1 and standard deviation 0.26. 0 is the lowest possible
risk (a few companies-very established, consistent
performers-have b very near
0); technically a risk can be as high as possible- in
reality the maximum is near 5.
| a) |
Identify the response variable
being measured. Is it categorical or
quantitative? |
| b) |
Identify the population and all
parameters. |
| c) |
How can the average risk be 1
when the minimum is 0 and the maximum is 5? |
| d) |
Take a simple random sample of
one stock and measure its b
index. With the information given is it possible
to compute P[X > 1.15]? Why (not)? If not, how
can this probability be found? |
| e) |
Suppose 20 stocks are selected
by simple random sample from all listed stocks.
Let X-bar be the average risk for the 20. Would
you consider X-bar a parameter or a statistic? |
| f) |
What are the mean and standard
deviation of X-bar's distribution? |
| g) |
Use the Central Limit Theorem
(assume n = 20 is sufficiently large) to
approximate P[X-bar > 1.15]. |
| h) |
My stock portfolio contains 20
stocks with (sample) mean b
index of 1.151. Using the previous part's answer,
what can you say about my portfolio? |
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