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Lab Activity 4:
Statistics, Parameters, and Sampling Distributions
Part I – Statistics and Parameters
In each situation, explain whether the value given in bold print is a statistic or a parameter:
1. A polling organization samples 1000 adults nationwide and finds that 72% of those sampled favor
tougher penalties for persons convicted of drunk driving.
2. In their year 2000 census, the United States Census Bureau found that the median age of all
American citizens was about 35 years.
3. For a sample of 20 men and 25 women, there is a 14 centimeter difference in the mean heights of the
men and women.
4. A writer wants to know how many typing mistakes there are in her manuscript, so she hires a
proofreader who tells her that in all books of the same length as hers, the average is 15 errors.
Explain whether p̂ or p is the correct statistical notation for each proportion described:
5. The proportion that smokes in a randomly selected sample of n = 300 students in the 11th and 12th
grades.
6. The proportion that smokes among all students in the 11th or 12th grade in the United States.
7. The proportion that is left-handed in a sample of n = 250 individuals.
Part II- Sampling Distribution of Sample Proportion:
A newspaper conducts a poll to determine the proportion of adults who favor a certain candidate.
They ask a random sample of 400 people whether or not they favor that candidate (Assume no bias!).
Suppose the true proportion of adults who favor the candidate is 64%.
1. The newspaper records the sample proportion who favor the candidate. What is the approximate
sampling distribution of the sample proportion? (i.e., What is the name of the distribution, and what
are its mean and standard deviation?) Hint: Look at the bottom of the slide “Sampling Distribution
of p-hat”
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2. What is the probability that the newspaper would have recorded a sample proportion greater than
68%?
3. What is the probability that less than 55% of the newspaper respondents would support this
candidate?
Part III- Sampling Distribution of Sample Mean:
Put the CD that came with your textbook into the CD-ROM drive, the menu for the CD should appear on the
screen automatically. Click on “Turn on Your Computer Applets”, then “Load Browser with Applets Index”.
Select the “SampleMeans” applet. This simulation will aid us in understanding this difficult concept of the
sampling distribution for a sample mean.
1. As you can see, we are looking at a population that is normally distributed with μ = 8 and σ = 5.
Let’s say that we want to take a sample of size 10 from this population (n = 10). To do this, enter
“10” in the box at the top of the applet labeled “# observations per sample”, then press the button
marked “1” under “# samples”. This will take one (1) sample of size 10 from the population. We
can see the actual elements of the sample listed in the box at the bottom of the applet, to the right of
the label “Current Sample:”. The applet also computes the mean of this sample ( x ) and displays it
in the box on the right of the applet, under the label “Sample Means.” There should also be a red bar
somewhere on the graph of the distribution of sample means. This histogram corresponds to the
values of x computed from the different samples we take.
What was the mean of the sample you took? Now, press the button marked “1” again. This will take
another sample of size 10 from the same population, and again calculate the new sample’s mean and add it to
the histogram.
What is the mean corresponding to the second sample?
Is it the same as the first sample’s mean? Should it be?
The histogram in red at the bottom of the applet is the histogram of the sampling distribution of x .
Now press the “Clear” button.
2. Now we’re back to the beginning, still looking at a population that is normally distributed with μ = 8
and σ = 5. Let’s say that we want to take a sample of size 10 from this population (n = 10). What
does the sampling distribution of x look like in this case? You can get an idea by using this
simulation.
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Enter “10” in the box at the top labeled “# Observations per sample”. Now click on the button
labeled “100” under “# Samples”. This will generate 100 samples of size 10 (n=10) from the
population, calculate the sample mean for each, and create a red histogram that shows the
distribution of those 100 sample means.
Click on the “100” button again. Roughly what shape is the histogram for these 200 sample means?
What are the center and spread? Does the spread make sense?
3. From the CD menu, click “Turn on Your Computer Applets” again and choose “SampleMeans”
once more. This should open a new browser window with another applet. KEEP the previous one
as well for comparison purposes. (Note: You may have to first open a new web browser before
going back to the CD menu in order to get both browser windows at the same time.)
Suppose we now want to take a sample of size 100. What would the sampling distribution of x look
like in this case? This time, enter “100” as the number of observations per sample and then generate
100 samples.
As you increased the sample size, what changed about the sampling distribution for x ? Has the
overall shape changed? the mean? the spread (variability)?
4. Instead of just looking at a histogram, let’s actually make some calculations in order to describe
these sampling distributions. Using the normal curve approximation rule (you should be able to find
this in the slides or in the book), describe the sampling distribution of x for samples of size 10 from
the population above. What is the shape? mean? standard deviation?
5. Now describe the sampling distribution of x for samples of size 100. What is the shape?
mean? standard deviation? Compare these results to your answer in #4. Do you notice the same
similarities/differences here as you did using the simulation?
Suppose I take a sample of size 100 from the population above. What is the probability that I will
observe a sample mean greater than 9? (Hint: You know how to do this calculation – think Zscores!)
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6. Remember that the population does not have to be normally distributed in order for the normal
approximation rule to apply, but our sample size does have to be relatively large (n ≥ 30). To
illustrate this, open the “TVMeans” applet.
Notice that the population is now quite skewed to the right. However, use the simulation to generate
100 samples of size 5 (n = # observations per sample = 5) from this population and look at the
distribution of the sample means. What do you notice about the shape?
7. Now clear the applet and generate 100 samples of size 30 (n=30) from this population. What is the
shape of the sampling distribution now?
Calculate the mean and standard deviation for the sampling distribution of x in this case.
If I took a sample of size n=30 from this population and calculated the sample mean of hours
watching TV in a week, what is the probability the sample mean would be between 8 and 9 hours?
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