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Chapter 18 – Part I
Sampling Distribution for Sample Proportion Statistic
Review
Categorical Variable One Category vs. All Other Categories
Examples
What political party is the U.S. senator?Ex. Democrat, All Other Parties
Who will you vote for in 2008 election?Hillary Clinton, All Other Candidates
What color are your eyes?Blue, All Other Eye Colors
Review
Population characteristicsp = proportion of population members in One
Category1-p = proportion of population members in All
Other Categories
Examples
p = proportion of U.S. senators who are Democrats.
p = proportion of voters who will vote for Hillary Clinton.
p = proportion of people with blue eyes.
Review
SRS characteristics_____ = proportion of sample members in
One Category_____ = proportion of sample members in All
Other Categories
Examples
_____ = proportion of U.S. senators in sample of size 10 who are Democrats.
_____ = proportion of voters in sample of size 2000 who will vote for Hillary Clinton.
_____ = proportion of people in sample of size 5 with blue eyes.
Randomness of Sampling
Random event = ___________________.
Before taking sample ________________.
Repeated Sampling
Repeat taking SRS of size n.Example: 2002 Senate.______ = proportion of Democratic Senators
in sample of size _______.
What happens to values of ________?
Repeated Sampling by Hand
Repeated Sampling Using Computer
Sampling Distribution
Many, many possible samples. Many, many possible values of ________. These ___________ are DATA.
Summarize DATA values!!
Mean (Center)
pp )ˆ(
Standard Deviation (Spread)
n
ppp
)1()ˆ(
Example
50% of registered voters plan to vote for Hillary Clinton. Select n people. (2, 5, 10, 25) Find sample proportion of registered voters planning to
vote for Clinton. Repeat sampling. What does sampling distribution of sample proportion
look like?
Example
10% of all people are left handed. Select n people. (2, 10, 50, 100) Find sample proportion of left handed people. Repeat sampling. What does sampling distribution of sample
proportion look like?
Shape
Two conditions1. np and n(1-p) are both larger than 10.
2. sample size n is less than 10% of population size.
Shape is approximately a ________________________!!!
Sampling Distribution for _____ Check for two conditions
Example
U.S. Senators Check assumptions (p = 0.54)
1. 10(0.54) = 5.4 and 10(0.46) = 4.6
2. n = 10 is 10% of the population size. Assumption 1 does not hold. Sampling Distribution of ___________
Example #1
Public health statistics indicate that 26.4% of the adult U.S. population smoke cigarettes. Describe the sampling distribution for the sample proportion of smokers among a random sample of 50 adults.
Example #1 (cont.)
Check assumptions
Example #1 (cont.)
Example #2
A student in Statistics 101 flipped a fair coin 200 times. What is the sampling distribution of the proportion of heads flipped?
Example #2 (cont.)
Check assumptions
Example #2 (cont.)
Example #3
Information on a packet containing 160 seeds that the germination rate of the seeds is 92%. Assume the seeds are a random collection of all seeds produced. What is the sampling distribution of the proportion of seeds in the packet that will germinate?
Example #3 (cont.)
Check assumptions
Example #3 (cont.)
68-95-99.7 Rule
For all normal distributionsApprox. 68% of all observations are between
_________ and __________.Approx. 95% of all observations are between
_________ and __________.Approx. 99.7% of all observations are
between _________ and __________.
68-95-99.7 Rule for Sampling Distribution “all observations” – What does this mean?
68-95-99.7 Rule – Example #1
68-95-99.7 Rule – Example #1
68-95-99.7 Rule – Example #1
68-95-99.7 Rule – Example #2
68-95-99.7 Rule – Example #2
68-95-99.7 Rule – Example #2
68-95-99.7 Rule – Example #3
68-95-99.7 Rule – Example #3
68-95-99.7 Rule – Example #3
Using the Sampling Distribution
Using the 68-95-99.7 rule, we know what to expect for the values of _______ for a given value of ________.
Can we use the normal distribution to find the probability of Having a __________ value below a certain amount? Having a __________ value above a certain amount?
Example #1
What is the probability of getting a random sample of 50 adults with 18 or more smokers?
Example #1
Example #2
What is the probability that in flipping a fair coin 200 times, we would get 80 or fewer flips with heads?
Example #2
Example #3
What is the probability that in a packet of 160 seeds we would see 138 or fewer seeds germinate?
Example #3