Chapters 8 & 9 lab.
Due: Tuesday, April 8.
Lab Instructional Video - http://www.youtube.com/watch?v=Mz_wbgN1J_0
This week’s lab will have two main components. First, you will explore the distribution of sample
means. The essential idea here is that the larger the n for each sample, the smaller the standard
deviation of the sample means and the more the distribution of the sample means is likely to be
approximately normally distributed. The second part of the lab will have you work with confidence
intervals for a population proportion.
1. Go to MyStatLab’s Sampling Distributions applet via their “Tools for Success”:
http://media.pearsoncmg.com/aw/aw_mml_shared_1/statistics/West_Applets/sampledist.html
Choose the Skewed distribution for the population. Note the original mean for this population
and standard deviation for this right skewed population. (A mean of about 15.5 and and a
standard deviation of about 12.5.) Hitting the “Sample” button will create N samples each of
size n. You can get a histogram of the sample means of these means if you have the box set to
“Mean”. The histogram of those sample means are displayed as well as their mean.
a. Fill in the below table in your lab document using N= 10,000 simulations for each sample
size n. Use the output that is displayed as “Sample Means” (third graph from the top).
(It may take a little while for the data to display after you click ‘Sample’ when
you get to larger sample sizes )
From the first graph: Population mean=________
From the first graph: Population sd = σ=_______
Size of
each
sample
Mean of the
10,000 sample
means
Sample sd of the
10,000 sample
means
(should be near 15.5)
(should be near 12.5)
𝜎
√𝑛
= standard error of
sample means
(Should be close to preceding
column’s value. Use σ from
above. )
n=2:
n=3:
n=5:
n=10:
n=30:
n=50:
n=100:
n=1000:
n=4000:
b. At what value of n does the graph of the sample means begin to look normal and have no
sign of right skewness?
c. What happens to the standard deviation of the means as n increases in size?
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2.
From the MyStatLab java applets menu, choose “Confidence Intervals for a Mean (the
Impact of Confidence Level)”. The url is
http://media.pearsoncmg.com/aw/aw_mml_shared_1/statistics/West_Applets/meanci.html . Click
the “Simulate” button, which will generate 100 samples of size n. Each green segment of a
horizontal bar represents a 95% confidence interval. The blue tips of each horizontal bar show the
extent of the 99% confidence interval. If the entire horizontal bar is red, then the 99% (and thus
95%) confidence interval did not contain the population mean value of 50. If only the middle part
of the horizontal bar is red (not green), then the 95% confidence interval did not contain the
population mean of 50, but the 99% confidence interval did contain 50.
a.
Click the “Clear” button. Set n=10, Distribution=Normal, Mean=50 and Std.
Dev.=10. Perform 1,000 simulations by hitting the ‘Simulate’ button 10 times. (The number for
“Contained mean” and “Did not contain mean” should add to 1,000.) What proportion of the 95%
Confidence Intervals contained the mean of 50? What proportion of the 99% confidence intervals
contained the mean of 50?
b.
Click the “Clear” button. Set n=100, Distribution=Normal, Mean=50 and Std.
Dev.=10. Perform 1,000 simulations. What proportion of the 95% Confidence Intervals contained
the mean of 50? What proportion of the 99% confidence intervals contained the mean 50?
c.
Click the “Clear” button. Set n=10, Distribution=Right Skewed, Mean=50 and Std.
Dev.=10. Perform 1,000 simulations. What proportion of the 95% Confidence Intervals contained
the mean of 50? What proportion of the 99% confidence intervals contained the mean of 50?
d.
Click the “Clear” button. Set n=100, Distribution=Right Skewed, Mean=50 and Std.
Dev.=10. Perform 1,000 simulations. What proportion of the 95% Confidence Intervals contained
the mean of 50? What proportion of the 99% confidence intervals contained the mean of 50?
e.
Your results for 2a and 2b should both be near 0.95 and 0.99. Your results for 2c will be a
few hundredths away 0.95 and 0.99 while 2d will be close to 0.95 and 0.99. Why do the confidence
intervals work well for normal distribution regardless of sample size, but only work well for the right
skewed distribution when the sample size is large (n=100)? (Hint: CLT)
3.
In this problem you will simulate how a population of 25,000 people will vote on a ballot
initiative and a voter opinion survey of 400 people randomly sampled from the population. First,
let’s create the population where about 55% of the people will vote “Yes” which we will denote as a
1, while the others will vote “No” as denoted by a 0.
In StatCrunch do the following: Data > Simulate > Bernoulli > Rows=25000, Columns=1, p=0.55,
click “Simulate”. You should now have a column of 0’s and 1’s representing the “Yes” and “No”
votes. (A Bernoulli random variable is equivalent to a binomial random variable with n=1.)
To see what the election results will be, Stat > Tables > Frequency > Select Columns=Bernoulli1 >
click “Calculate”. This represents the proportion of your simulated population that will vote yes.
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Next, we do the random survey of 400 people. In StatCrunch: Data > Sample Columns > Select
Columns(s)=Bernoulli1, Sample size=400, Number of samples=1, click “Sample Columns(s)”. Now to
summarize the survey data: Stat > Proportions > One sample > with data > Outcomes in=
Sample(Bernoulli1), Success=1, click “Next” > select “Confidence Interval”, Level=0.95,
Method=Standard-Wald, click “Calculate”.
i.
What was your relative frequency of 1’s in the population of N=25,000?
ii.
What was your relative frequence of 1’s in your sample of n=400?
iii.
List out your lower (L.Limit) and upper (U.Limit) bounds of your 95% confidence interval.
iv.
Did your interval contain the true proportion that you calculated for part i?
v.
About what percent of the students in this class will contain their population proportion
(population relative frequency) in their 95% confidence interval?
vi.
What is your margin of error for your sample? (Hint: Margin of error, not standard error.)
4. Find an online article that does a survey that involves binomial (yes, no) data. Make sure that the
article states the margin of error and the sample size - often it’s in small print or at the very end of
the article. State your news source and URL (web address) , the date, and article title, the topic
surveyed, the surveyed percent, margin of error and sample size. Using their sample size, can you
duplicate their margin of error?
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