### 7.2 Generating Sampling Distributions

Warm-up
7.2 Generating Sampling Distributions
(C) X is close to the mean and so will have a z-score close
to 0. Boxplots, if they show an isolated point, will show only
outliers. X and the two clusters are clearly visible in a
stemplot of these data.
School of the day!
Block 4
School of the day!
Block 5
Answers to H.W. E #4 and 5
7.2 Generating Sampling Distributions
Sampling Distribution of a Mean
If you toss a fair die 10,000 what do you expect the histogram
of the results to look like?
How would you expect the histogram to look like if you were
graphing the results of the average of two die?
Central Limit Theorem
The sampling distribution of any mean becomes more
nearly Normal as the sample size grows. Most
importantly the observations need to be independent
and collected with randomization.
FYI “central” in the theorem name means “fundamental”
CLT and Equations
• The CLT requires essentially the same assumptions and
conditions from modeling proportions: Independence,
Sample size, Randomization, 10% and large enough sample
The standard deviation of the sampling distribution is
sometimes called the standard error of the mean.
x 

n
standard
dev. of pop. / sample size
Properties of Sampling Distribution of Sample Mean
pg 430
If a random sample of size n is selected from a population
with mean  and standard deviation , then
• The mean  x of the sampling distribution of x equation
equals the mean of the population,  :
• The standard deviation  x , the sampling distribution of
x , sometimes called the standard error of the mean:
x 

standard
dev. of pop. / sample size
n
• Using the formula above you can find standard error of
the sample mean without simulation
Number of Children Problem
Physical Education Department and BMI study
A college physical education department asked a random sample of 200
female students to self-report their heights and weights, but the
percentage of students with body mass indexes over 25 seemed
suspiciously low. One possible explanation may be that the respondents
“shaded“ their weights down a bit. The CDC reports that the mean
weight of 18-year-old women is 143.74 lb, with a standard deviation of
51.54 lb, but these 200 randomly selected women reported a mean
weight of only 140 lb.
Question: Based on the Central Limit Theorem and the 68-95-99.7 Rule,
does the mean weight in this sample seem exceptionally low or might
this just be random sample-to-sample variation?
Common Mistakes on Test
what is the mean and standard deviation for the number of defects for pairs of shoes produced by this company.
Form A
1. This table gives the percentage of women who ultimately have a
given number of children. For example, 19% of women ultimately
have 3 children. What is the probability that two randomly selected
women will have a combined total of exactly 2 children?
0 and 2,
1 and 1,
2 and 0
0.18* 0.35 + 0.17*0.17+ 0.35* 0.18 = 0.159
Form B 2. For the sake of efficiency, a shoe company decides to produce
the left shoe of each pair at one site and the right shoe at a
different site. If the two sites produce shoes with a number of
defects reflected by  1  0 . 002 ,  1  0 . 15 and  2  0 . 005 ,  2  0 . 18 ,
 X Y   X   Y
0 . 002  0 . 005  0 . 007
 X  Y   X   Y 0 . 15
2
2
2
2
 0 . 18
2

0 . 0549  . 234
Expected Number of Success and Expected Number of Trials
Binomial Distribution: Flipping a coin 6 times, about how
many times flip head on average.   n p  6 ( 0 . 5 )  3
x
In a simple random sample of 15 students, how many are expected to be
younger than 20 . 60% of students are under 20.
Geometric Distribution: Flipping a coin, when do you expect
1
(on average) to have your first success.
x 
p
What is the expected number of interview before the second person
without health insurance is found? (16% no health insurance)
Expected Number of Success and Expected Number of Trials
Binomial Distribution: Flipping a coin 6 times, about how
many times flip head on average.   n p  6 ( 0 . 5 )  3
x
What is the average number of students without laptops you would
expect to find after sampling 5 random students? (60% w/ laptops)
Geometric Distribution: Flipping a coin, when do you expect
(on average) to have your first success.
1
x 
p
On average how many otters would biologists have to check before
finding an infected otter? (20% are infected)