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Rockford High School
AP Statistics
Chapter 9 Note Packet: Sampling Distributions
Mrs. Shutich
2012-2013
Sampling Distribution of Rockford ACT Scores Simulation
ACT scores are normally distributed with
μ=
σ=
Simulate a sample of 100 and record sample mean & sample standard deviation.
Repeat two more times.
Sampling Distribution of Rolling a Dice Simulation
Outcomes of rolling a dice are uniformly distributed with
μ=
σ=
Simulate a sample of 50 rolls and record the sample mean and sample standard deviation. Repeat two more times.
9.1 Sampling Distributions
_______________: A number that describes the population. μ, σ, p
_______________: A number that describes a sample- used to estimate an unknown population
̅, s, p
parameter. 𝒙
Read Ex. 9.1 on page 564:
Do Ex. 9.2 on page 565: A Gallop poll asked a random sample of 515 U.S. adults whether they
believe in ghosts. 160 said "yes'. Find the sample proportion that say they believe in ghosts.
Sampling Variability: What would happen if we took many samples?
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


Do Ex. 9.3 on page 565: At the Custom Inspection Area in Guadalajara airport passengers who
receive a red light are searched. Customs claims that the probability of a green light is 0.70. Let’s
take a look at a simulation of 1000 SRSs of 100 travelers.
The ________________________ distribution of a statistic is the distribution of values taken by the
statistic in all possible samples of the same size from the same population.
Read Ex. 9.4 on page 567: Consider the random digits table probability distribution. Now consider
the process of taking an SRS of size n = 2 and computing the 𝑥̅ for the sample and repeating this
process 100 times. This is now a sampling distribution of 𝑥̅ .
What do you notice about the sampling distribution of 𝑥̅ for
samples of size n = 2? See figure 9.4.
Describing Sampling Distributions
1.
2.
3.
4.
Describe the sampling distribution in Ex. 9.5 on page 571:
According to 2005 Nielsen ratings, Survivor: Guatemala was one of the most -watched television
shows in the U.S. during every week that it aired. Suppose that the true proportion of U.S. adults who
watched is p = 0.37. Figure 9.5 shows a simulation of 1000 SRS's of size n=100 from the population.
The Bias of a Statistic: Sampling distributions allow us to describe bias more precisely (bias of a
statistic rather than sampling method). Bias concerns the center.
________________ Statistic: A statistic used to estimate a parameter that is unbiased where the
mean of its sampling distribution equals the true value of the parameter.
Read page 574.
Does ex. 9.5 show bias?
________________ of a Statistic is described by the spread of its sampling distribution. This
spread is determined by the sampling design and the size of the sample. Larger samples give
smaller spread. As long as population is at least 10 times as large as sample, the spread of the
sampling distribution is ≈ the same for any population size.
Do Ex. 9.6 on page 575: For ex. 9.5 (Survivor show ratings), between what two values will 95% of
the values of 𝑝̂ fall. (for n=100 and n=1000)
Homework: p.568 #9.1-9.3, 9.5
p.577 #9.8-9.10
9.2 Sample Proportions
Sampling Distribution of a Sample Proportion (𝑝̂ ): Suppose that 𝑝̂ is the sample proportion of an
SRS of size n from a large population with population proportion p.
𝝁𝒑̂ = _______________
𝝈𝒑̂ = _______________
Rule of Thumb 1: use the recipe for the standard deviation of 𝑝̂ only when _________________
Rule of Thumb 2: use the Normal approximation to the sampling distribution of 𝑝̂ for values of
n and p that satisfy __________________ and __________________.
Do Ex. 9.7 on page 584: A polling organization asks an SRS of 1500 first-year college students
whether they applied for admission to any other college. In fact, 35% of all first-year students applied
to colleges besides the one they are attending.
What is the probability that the random sample of 1500 students will give a result within 2% points of
this true value?
Do Ex. 9.8 on page 586: About 11% of American adults are black.
If a national sample of 1500 adults contains only 9.2% blacks, should we suspect that the sampling
procedure is somehow under representing blacks?
Homework: p.588 #9.19-9.22
9.3 Sample Means
Sampling Distribution of a Sample Mean (𝑥̅ ): Suppose that 𝑥̅ is the mean of an SRS of size n from
a large population with mean 𝜇 and standard deviation 𝜎.
𝝁𝒙̅ = ___________
𝝈̅𝒙 = ___________
Behavior of 𝑥̅ :
1.
2.
3.
Sampling Distribution of a Sample Mean from a Normal Population: Draw an SRS of size n from
a population that has the Normal distribution with mean μ and standard deviation σ. Then the sample
mean 𝑥̅ has the Normal distribution with mean μ and standard deviation 𝜎⁄ .
√𝑛
Do Ex. 9.10 on page 593: The population distribution of heights of young women is N(64.5, 2.5).
Find the mean and standard deviation of the Sampling Distribution of 𝑥̅ of an SRS of 10 young
women.
What is the probability that a randomly selected young women is taller than 66.5 inches?
What is the probability that the mean height of an SRS of 10 young women is greater than 66.5
inches?
Homework: p.595 #9.31-9.34
Central Limit Theorem
Draw an SRS of size n from any population (normal or not) with mean μ and finite standard deviation
σ. When n is large, the sampling distribution of 𝑥̅ is close to
______________________.
Do Ex. 9.13 on page 599: Suppose the time it takes a technician to perform preventive maintenance
on an air conditioning unit is μ = 1 hour and σ = 1 hour.
Your company has a contract to maintain 70 of these units in an apartment building. You must
schedule technicians/time for a visit to this building. Is it safe to budget an average of 1.1 hours for
each unit? Or should you budget an average of 1.25 hours?
Homework: p. 9.35, 9.37-9.39
Review: p.607 # 9.47-9.53 odds, 9.57.