Sampling Distributions
Review
Random phenomenon• Individual outcomes unpredictable
Sample space• all possible outcomes
Probability of an outcome• long-run proportion for outcome
Probability distribution• probabilities for outcomes in sample space
Review
parameter numerical fact about the population (e.g. m) – the
thing we want to know, but can’t
statistic corresponding numerical fact in the sample (e.g. ) – the thing we can know
x
Fact about (when X has mean μ and s.d. σ)
• Law of Large Numbers– As n gets larger, “gets closer and closer” to the
mean μ– More precisely, the chance of getting a “bad”
gets smaller as n gets larger
x
x
x
Sampling Distribution• A mental picture: imagine trying to estimate
average height in this class (μ) – A sample of 5 persons is obtained and is
calculated – Imagine all possible samples of size 5, with an
for each sample– Collect all the ’s: This is the “sampling
distribution of ”• A definition: The sampling distribution of a
statistic is the distribution of values taken by the statistic in all possible samples of size n
x
x
xx
What we did….Histogram of Heights: = 68.18; = 4.49
Height (inches)
# o
f stu
de
nts
60 65 70 75 80
02
04
06
08
0
Histogram of xbars: ̂x = 68.39; ̂x = 2.17
x (inches)
# o
f x v
alu
es
60 65 70 75 80
02
04
06
08
0 ̂x
Application to Statistics
if you have a statistic calculated from a random sample or randomized experiment
• sample space = all possible values of sample statistic
• The probability distribution of the sample statistic is called the sampling distribution
iClickerConsider a simple random sample of 100 BYU students, asking them how many movies they watched last week (x), and then calculating . What is the sampling distribution?
a. dist. of x for all BYU stud.b. dist. of x for 100 BYU stud.c. prob. of getting the
particular of the sampled. prob. dist. of for samples
of 100 BYU students
x
xx
Why sampling distribution?
• sampling distribution allows us to assess uncertainty of sample results (i.e., “how reliable is ?”)
• if we knew the spread of the sampling distribution, we would know how far our might be from the true m
x
x
Height Data for Our Class
• μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches• What is sampling distribution for if n=5?
– We can’t see the (theoretical) sampling distribution because we don’t have time to look at all possible samples of size 5
– We CAN approximate it with simulation• How does the sampling distribution of
compare with distribution of heights (x)?
x
x
What we did….Histogram of Heights: = 68.18; = 4.49
Height (inches)
# o
f stu
de
nts
60 65 70 75 80
02
04
06
08
0
Histogram of xbars: ̂x = 68.39; ̂x = 2.17
x (inches)
# o
f x v
alu
es
60 65 70 75 80
02
04
06
08
0 ̂x
If we had truly random samples….Histogram of Heights: = 68.18; = 4.49
Height (inches)
# of
stu
dent
s
60 65 70 75 80
010
2030
40
Histogram of xbars when n= 5 : mean of xbars = 68.23 (should be 68.18)
sd of xbars = 1.98 (should be 4.49/sqrt(n)= 2.01 )
xbars
Den
sity
60 65 70 75 80
0.00
0.05
0.10
0.15
0.20
x
Histogram of xbars when n= 16 : mean of xbars = 68.16 (should be 68.18)
sd of xbars = 1.08 (should be 4.49/sqrt(n)= 1.12 )
xbars
Den
sity
60 65 70 75 80
0.0
0.1
0.2
0.3
x
Histogram of xbars when n= 100 : mean of xbars = 68.14 (should be 68.18)
sd of xbars = 0.39 (should be 4.49/sqrt(n)= 0.45 )
xbars
Den
sity
60 65 70 75 80
0.0
0.2
0.4
0.6
0.8
1.0
x
More facts about (when X has mean μ and s.d. σ)
• Sampling Distribution (aka “Theoretical Sampling Distribution”) for – Has a mean of exactly μ– Has a standard deviation of exactly
x
x
n
Height Data for Our Class
• μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches• Someone says BYU’s incoming class for Fall
2014 will have a mean height larger than 68.18, based on a random sample of n=5 incoming freshman with = 69.5. What do you think?– What if the came from a sample with n=16?
n=100?x
x
Height Data for Our Class
• μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches• QUIZ: What is the mean of the sampling
distribution for if n=4?– A: impossible to know– B: exactly 68.18 inches– C: approximately 68.18 inches, give or take a little
bit of room for error– D: a value that gets closer and closer to 68.18
inches as n gets larger and larger
x
Height Data for Our Class
• μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches• QUIZ: What is the standard deviation of the
sampling distribution for if n=4?– A: impossible to know– B: 4.49 inches– C: 4.49/2 = 2.245 inches– D: 4.49/4 = 1.1225 inches
x
sampling distribution applet
T F1. always estimates μ well (i.e., it’s
always close to μ). 2. Using the sampling dist. we can
compute probabilities on .3. does not vary from sample to
sample.4. The mean of the sampling dist. of
is µ.
iClicker
x
xx
x
Next…• What if we don’t have the whole population to
simulate from?• What if we don’t have 600 Stat 121 students
willing to calculate values based on 600 different samples?– What if we only have time for one sample of size
n=35 (BYU students), and we get 6.9 hours as an average number of TV hours per week? Can we say that BYU students’ mean viewing time is significantly less than the national average of 10.6 hours for college students? (σ=8.0) What if knew somehow that the sampling distribution for is normal?
x
x
Vocabulary
StatisticParameterProbabilityProbability distributionSampling distribution of statistic
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