AP Statistics
Chapter 7 Outline
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7.1 Sampling Distribution
#1-13 odd, 17-20
7.2 𝑝̂
21-24, 27, 29, 33, 35, 37, 41
7.3 𝑥̅
43-46, 49, 51, 53, 55, 57, 59, 61
Chapter 7 Review
Chapter 7 Test
Semester Review
Final Exam
7.1 What is a sampling distribution?
Parameter
Statistic
Sampling Distribution
Unbiased Estimator
The variability of a statistic is described by the spread of its sampling distribution.

This spread is determined primarily by the


__________give smaller spread.
The spread of the sampling distribution does not depend on the size of the population, as long as
the population is at least
𝝁:
̅:
𝒙
𝒑:
̂:
𝒑
_____ of the random sample.
_______________
̂:
7.2 The Sampling Distribution of a Sample Proportion 𝒑
Choose a SRS size n from a population size N with proportion p of successes.
𝑝̂ : sample proportion of successes

The mean of the sampling distribution of 𝑝̂ is:

The standard deviation of the sampling distribution of 𝑝̂ is:

As long as 10% condition is satisfied:

As n increases, the sampling distribution of 𝑝̂ becomes approximately Normal.

Check Normal conditions:
̅
7.3 The Sampling Distribution of a Sample Mean 𝒙
Suppose that 𝑥̅ is the mean of an SRS of size n drawn from large population with mean 𝜇 and standard
deviation 𝜎

The mean of the sampling distribution of 𝑥̅ is:

The standard deviation of the sampling distribution of 𝑥̅ is:

As long as 10% condition is satisfied:

If the population distribution is Normal, then so is 𝑥̅

If the population distribution is not Normal, 𝑥̅ will be approximately Normal if:
Sample Proportion Problem 𝑝̂ :
About 75% of young adult Internet users watch online video. Suppose that a sample survey consists of
1000 young adult Internet users and calculates the proportion 𝑝̂ in this sample who watch online video.
a)
What is the mean of the sampling distribution?
b)
Find the standard deviation of the sampling distribution of 𝑝̂
c)
Is the sampling distribution of 𝑝̂ approximately Normal? Check Normal Conditions.
d)
Is the sample size were 9000 rather than 1000, how would this change the sampling distribution
of 𝑝̂
Sample Proportion Problem 𝑝̂ :
A polling organization asks an SRS of 1500 first-year college students how far away their home is. Suppose
that 35% of all first-year students actually attend college within 50 miles of home. What is the probability
that the random sample of 1500 students will give a result within 2 percentage points of this true value?
Sample Mean Problem 𝑥̅
Suppose that the number of movies viewed in the last year by high school students has an average of 19.3
with a standard deviation of 15.8. Suppose we take an SRS of 100 high school students and calculate the
mean number of movies viewed by the members of the sample.
a) What is the mean of the sampling distribution of x ?
b) What is the standard deviation of the sampling distribution of x ? Check whether the 10% condition
is satisfied.
Sample Mean Problem 𝑥̅
The height of young women follows a Normal distribution with mean µ = 64.5 inches and standard
deviation σ = 2.5 inches.
a) Find the probability that a randomly selected young woman is taller than 66.5 inches.
b) Find the probability that the mean height of an SRS of 10 young women exceeds 66.5 inches.
Sample Mean Problem 𝑥̅
The length of human pregnancies from conception to birth varies according to a distribution that is approximately
Normal with mean 266 days and standard deviation 16 days.
a) Find the probability that a randomly selected chosen pregnant woman has a pregnancy that lasts for more than
270 days.
b) Suppose we choose an SRS of 6 pregnant women. Let 𝑥̅ = the mean pregnancy length for the sample. What is
the mean of the sampling distribution 𝑥̅ ?
c) What is the standard deviation of the sampling distribution 𝑥̅ . Check the 10% condition.
d) Find the probability that the mean pregnancy length for the women in the sample exceeds 270 days.
Sample Mean Problem 𝑥̅
Suppose that the number of texts sent during a typical day by a randomly selected high school student
follows a right-skewed distribution with a mean of 15 and a standard deviation of 35. Assuming that
students at your school are “typical texters”, how likely is it that a random sample of 50 students will have
sent more than a total of 1000 texts in the last 24 hours?