![Page 1: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/1.jpg)
LECTURE 15THURSDAY, 26 MARCH
STA 291Spring 2009
![Page 2: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/2.jpg)
Le Menu• 9 Sampling Distributions
9.1 Sampling Distribution of the Mean9.2 Sampling Distribution of the Proportion
Including the Central Limit Theorem (CLT), the most important result in statistics
• Homework Saturday at 11 p.m.
![Page 3: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/3.jpg)
Going in Reverse, S’More
• What about “non-standard” normal probabilities?
Forward process: x z probReverse process: prob z x
• Example exercises:p. 274, #8.35, 37; p. 275, #49
3
![Page 4: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/4.jpg)
Typical Normal Distribution Questions
• One of the following three is given, and you are supposed to calculate one of the remaining1. Probability (right-hand, left-hand, two-sided, middle)2. z-score3. Observation
• In converting between 1 and 2, you need Table 3.
• In transforming between 2 and 3, you need the mean and standard deviation
![Page 5: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/5.jpg)
Chapter 9 Points to Ponder
• Suggested Reading– Study Tools Chapter 9.1 and 9.2– OR: Sections 9.1 and 9.2 in the textbook
• Suggested problems from the textbook:9.1 – 9.4, 9.18, 9.30, 9.34
![Page 6: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/6.jpg)
Chapter 9: Sampling Distributions
X
Population with mean and standard deviation
X
X
XXXXX
X
• If you repeatedly take random samples andcalculate the sample mean each time, thedistribution of the sample mean follows apattern• This pattern is the sampling distribution
![Page 7: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/7.jpg)
Properties of the Sampling Distribution
Expected Value of the ’s: .
Standard deviation of the ’s:
also called the standard error of
(Biggie) Central Limit Theorem: As the sample size increases, the distribution of the ’s gets closer and closer to the normal.
X
Xn
X
X
Consequences…
![Page 8: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/8.jpg)
Example of Sampling Distribution of the Mean
As n increases, the variability decreases and the
normality (bell-shapedness) increases.
8
![Page 9: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/9.jpg)
Sampling Distribution: Part Deux
Binomial Population
with proportion p of successes
• If you repeatedly take random samples andcalculate the sample proportion each time, thedistribution of the sample proportion follows apattern
p̂p̂p̂p̂p̂p̂p̂p̂
p̂
![Page 10: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/10.jpg)
Properties of the Sampling Distribution
Expected Value of the ’s: p.
Standard deviation of the ’s:
also called the standard error of
(Biggie) Central Limit Theorem: As the sample size increases, the distribution of the ’s gets closer and closer to the normal.
Consequences…
p̂
p̂
p̂
p̂
n
pp 1
![Page 11: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/11.jpg)
Example of Sampling Distributionof the Sample Proportion
As n increases, the variability decreases and the
normality (bell-shapedness) increases.
11
![Page 12: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/12.jpg)
Central Limit Theorem
Thanks to the CLT …
We know is approximately
standard normal (for sufficiently large n, even if the original distribution is discrete, or skewed).
Ditto
n
X
npp
pp
1
ˆ
12
![Page 13: LECTURE 15 THURSDAY, 26 MARCH STA 291 Spring 2009](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649e9d5503460f94b9df6d/html5/thumbnails/13.jpg)
Attendance Question #16
Write your name and section number on your index card.
Today’s question:
13