Download Unit 4 B Schedule

DatStat UNIT 4-B Schedule (Topics 16-18)
Day 1:
________
Journal U4.B1
Topic 16: Sampling Distributions: Proportions
SWBAT: Understand how Sampling Distributions of Sample
Proportions( pˆ ), behave. Define the Central Limit Theorem for pˆ .
Activity: Reese’s Pieces
HW 3: p. 358 #16 – 6 , 7, 9 (a,b,c only)

Day 2:
________

Journal U4.B2
Topic 17: Sampling Distributions: Means
SWBAT: Understand how Sampling Distributions of Sample Means ( x ),
behave. Define the Central Limit Theorem for x .
Activity: Pennies
HW 4: p. 382 # 17-9,10


Day 4:
________
Journal U4.B3
Topic 18: The Central Limit Theorem
SWBAT: define the CLT and decide when the CLT applies to Sampling
Distributions of Sample Means and Sample Proportions.
HW 5: p 396 # 18-6, 7, 11
Day 6:
Journal U4.B4
REVIEW DAY
Day 7:
QUIZ UNIT 4B (Topics 16-18)
_______
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Topic 16: The Sampling Distribution of Proportions
Sampling Distribution: _____________________________________________(DUH!)
Value:
Population = Parameter
Sample = Statistic
Mean
Standard Deviation
Percent (Proportion)
Size
Activity 16-2: I will give you a Sample of 25 Reese’s Pieces candies from a Population
of the candies in a bag. I want you to make a count and proportion of each color. Only
then may you eat them!
Orange
Yellow
Brown
Count
Proportion(%)
a) Is the proportion of Orange candies in your sample a parameter or a statistic? _______
What symbol is used to denote it?______
b) Is the proportion of Orange candies in the teacher’s bag a parameter or a statistic? ___
What symbol is used to denote it?______
c) Do you know what the population proportion of Orange candies is from the bag? ____
d) How would you estimate that proportion?
e) We will build a distribution of the samples of the proportion of Orange candies.
0 .04 .08 .12 .16 .20 .24 .28 .32 .36 .40 .44 .48 .52 .56 .60 .64 .68 .72 .76 .80
f) How would you describe this SAMPLING DISTRIBUTION (this class SUCS)?
g) Between what two values do the middle 95% of all the samples fall?
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Sample Proportions:
1. Log onto the website: http://www.rossmanchance.com/applets/ Select “Reeses
Pieces”
2. Choose “Sample Size” to be 10.
3. Hit the “Draw Samples” button. How many candies were selected (n) ? ________ What
portion was orange ( )? __________. Repeat at least 5 times.
4. Undo the “Animate” box. Choose “Number of Samples” to be 10. Click “Draw Samples”
at least 20 times.
5. Describe what you see in the dot plot.
 Shape:________________________________________

Center:_______________________________________

Spread:_______________________________________

Unusual Points: _________________________________
6. Choose “Sample Size” to be 25. Select the “Animate” box. Choose “Number of Samples”
to be 1.
7. Hit the “Draw Samples” button. How many candies were selected (n) ? ________ What
portion was orange ( )? __________. Repeat at least 5 times.
8. Undo the “Animate” box. Choose “Number of Samples” to be 10. Click “Draw Samples”
at least 20 times.
9. Describe what you see in the dot plot.
 Shape:________________________________________

Center:_______________________________________

Spread:_______________________________________

Unusual Points: _________________________________
10. What did we learn about the sampling distributions of sample proportions as n gets
bigger? Talk about how shape, center, and spread changes.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
____________________________________________________
The Central Limit Theorem: If n is large (usually above 30), the sampling
distributions of the sample means and proportions are distributed Normally. They are
centered at µ and
p
respectively.
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CENTRAL LIMIT THEOREM (CLT) for SAMPLE PROPORTIONS: If we take a
SRS of size n from a large population, then the sampling distribution of the pˆ 's is
___________________, with
 pˆ  p
When does this happen?

 pˆ 
and
np ≥ 10
and
p(1 p)

n
n(1 – p ) ≥ 10

Ex1) Find the probability of getting a sample proportion of Orange candies of size 25
that is greater than 50%.
Ex2) Find the probability of getting a sample proportion of Orange candies of size 100
that is greater than 50%.
Ex3) Bonus Challenge (for the rest of the bag!): Without your calculators, predict the

standard deviation of the sampling distribution of the sample props pˆ , for n = 400 .

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Today, your goal is to learn about how SAMPLE MEANS behave. So what is a
sampling distribution? It’s a distribution of samples
Under score ( “Shift” then “ – “
Sample Means:
)
1) Log onto the website: http://onlinestatbook.com/stat_sim/sampling_dist/index.html
2) On the left hand side, click the “Begin” button.
3) What do you see? Note below: n = 5 .
 Describe the shape of the distribution:______________________________
 What is the mean of the “Parent Population”? ___________
 What statistical symbol should be used for the mean?_____________
 What is the Standard Deviation? ___________________
 What statistical symbol should be used for the std. deviation?_____________
4) Now hit the “Animated” button. What happened? Repeat at least 5 times.
5) What does the little blue “chunks” represent? __________________________________
6) Now click the “5” button. You are taking 5 samples at a time. Repeat at least 10 times
and watch it grow.
7) Now click the “1,000” button. You just took 1,000 samples from the population.
8) Compare the Blue Sampling Distribution with the Black Parent Population.
 What happened to Spread? __________________________________________
 What happened to the mean? ________________________________________
9) In the drop-down menu at the top right (it started with Normal), Choose “Skewed”.
Choose n=25 below for the “Distribution of Means”.
10) Now hit the “Animated” button. What happened? Repeat at least 3 times.
11) Now click the “5” button. You are taking 5 samples at a time. Repeat at least 10 times
and watch it grow.
12) Now click the “1,000” button. You just took 1,000 samples from the population.
13) Compare the Blue Sampling Distribution with the Black Parent Population.
 What happened to Spread? __________________________________________
 What happened to the mean? ________________________________________
14) Try clicking the “Fit Normal” box. What happened? Do you think sample means behave
normally?
15) In the drop-down menu at the top right (it started with Normal), Choose “Custom”.
Click and drag your cursor over the histogram and create your own distribution. Repeat
the steps above.
16) What can we conclude about the distribution of Sample Means? Talk about shape,
center, and spread in relation to the parent population.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
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Topic 17: Sample Means
Activity 17-1: PENNIES In this activity, you will be charting the age of pennies. For
example, how old is a penny from 2001? We will chart the ages from the entire
population. Then, I will ask you to start sampling from the population. We will look at
the sampling distribution of sample means ( x ) of size n = 10 . Make a rough sketch
of each distribution with a smooth curve
Population:
sampling distribution of ( x ) of size n = 10


a) Describe the shape of the population: ______________________________________
b) Guesstimate µ, the population mean: ______________________
c) Describe the shape of the SamplingDistribution: _____________________________
d) Now guesstimate µ, the population mean: ______________________ Which guess, b
or d, do you feel more confident in?____
e) Describe how the distribution changed (SUCS) from n = 1(the population) to n = 10
(the distribution of x ) .
Population (n = 1)
Sampling Dist. (n = 10)

Shape
Unusual Points
Center
Spread
CENTRAL LIMIT THEOREM (CLT) for SAMPLE MEANS: If we take a SRS of
size n from a large population, then the sampling distribution of the x ’s is
___________________, with
x  
When does this happen?

  x  
n
and
When
n ≥ 30

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Ex 1) Starbucks claims that a pour of a “Venti” coffee drink is Normally distributed and
will average 20 oz. They admit that there is some variation in pours and that the   .5oz
for their population of “Venti” coffees. What is the probability that my next coffee is less
than 19.3 oz.?

Ex2) If I take and measure 15 coffees, what is the probability that the sample mean of
these coffees is less than 19.3 oz.?
Ex 3) If I take and measure 30 coffees, what is the probability that the sample mean of
these coffees is less than 19.3 oz.?
Bonus: What statistical principal is being displayed by examples 1,2,3 (What happens to
the probabilities as n increases).
Describe in your own words what the CLT says about sample means and proportions
(Hint: Talk about how they are distributed and what happens to SPREAD as n increases).
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Topic 18: The CLT
Activity 18-1 (p. 387): The CDC (Center for Disease Control and Prevention) reported
that 22.9% of all American adults smoke regularly in 1998.
a) Is .229 a parameter or a statistic? What symbol should be used to denote it? ________
b) We want to see if that value has changed. Suppose you take a SRS of 100 American
adults and ask whether they smoke. What do we know about the distribution of this
sample proportion? (Talk about Shape, Center and Spread)
c) We found that 19 of our 100 adults were actually smokers. What is the probability
that we find a sample as low or lower than the one we obtained?
d) Interpret your answer from part (c). Does this constitute strong evidence that the
proportion of smokers is becoming smaller?
e) The CDC is encouraged by the results from ( c and d ). So now they hire you to
perform an extensive survey to determine if the proportion of smokers is actually
becoming smaller. They pay you to randomly sample 2500 American adults. You found
that 470 indicated that they smoke regularly. What is the probability that we find a
sample as low or lower than the one we obtained?
f) They now want you to estimate the true proportion of American adult smokers. You
want to be 95% sure that you create a window (min to a max) that will catch the true
proportion. What is your minimum and maximum estimates? (Hint: 68-95-99.7% Rule)
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Activity 18-3 (p.391) BigBar Candy Bar Company advertises that their famous BigBar
weighs 2.13 oz. Their website claims that the distribution of BigBar weights is Normal
with a mean of 2.2 oz and a standard deviation of .04 oz.
a) What is the probability that you purchase a BigBar and get cheated (less than the
advertised weight)?!
b) A box of BigBars contains 50 BigBars. What can we say about the distribution of the
sample mean weight of 50 BigBars?
c) What is the probability that you purchase a box of 50 BigBars and the average weight
is less than the advertised weight of 2.13?
d) Now draw the two distributions (1 single BigBar and the average of 50 BigBars) on
top of eachother. Shade the proportion that gets “cheated”.
e) What does your drawing indicate about the probability of a value getting far away
from the mean as sample size (n) increases?
f) Suppose your box of 50 BigBars had an average weight of 2.12 oz. What would you
say about the true weight of the BigBar?
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