bps - 3rd ed. chapter 101 sampling distributions
TRANSCRIPT
BPS - 3rd Ed. Chapter 10 1
Chapter 10
Sampling Distributions
BPS - 3rd Ed. Chapter 10 2
Parameters and Statistics Parameter
– Is a fixed number that describes the location or spread of a population
– Its value is NOT known in statistical practice Statistic
– A calculated number from data in the sample– Its value IS known in statistical practice– Some statistics are used to estimate parameters
Sampling variability– Different samples or experiments from the same
population yield different values of the statistic
BPS - 3rd Ed. Chapter 10 3
Parameters and statistics
The mean of a population is called µ this is a parameter
The mean of a sample is called “x-bar” this is a statistic
Illustration:– Average age of all SJSU students (µ) is 26.5– A SRS of 10 San Jose State students yields a
mean age (x-bar) of 22.3 – x-bar and µ are related but are not the same thing!
BPS - 3rd Ed. Chapter 10 4
The Law of Large Numbers
BPS - 3rd Ed. Chapter 10 5
Example 10.2 (p. 251)
Dimethyl sulfide (DMS) is sometimes present in wine causing off-odors
Different people have different thresholds for smelling DMS
Winemakers want to know the odor threshold that the human nose can detect.
Population values: – Mean threshold of all adults
is 25 µg/L wine – Standard deviation of all
adults is 7 µg/L – Distribution is Normal
Does This Wine Smell Bad?
BPS - 3rd Ed. Chapter 10 6
Simulation: Example 10.2 (cont.)
Suppose you take a simple random sample (SRS) from the population and the first individual in sample has a value of 28
The second individual has a value of 40 The average of the first two individuals =
(28+40)/2 = 34 Continue sampling individuals at random and
calculating means Plot the means
BPS - 3rd Ed. Chapter 10 7
Simulation: law of large numbers Fig 10.1
The sample mean gets close to population mean as we take more and more samples
BPS - 3rd Ed. Chapter 10 8
Sampling distribution of xbar
Key questions:
What would happen if we took many samples or did the experiment many times?
How would the statistics from these repeated samples vary?
BPS - 3rd Ed. Chapter 10 9
Case Study
• Recall = 25 µg / L, = 7 µg / L and the distribution is Normal
• Suppose you take 1,000 repetitions of samples, each of n =10 from this population• You calculate x-bar in each sample • You plot the x-bars as a histogram • You study the histogram (next slide)
Does This Wine Smell Bad?
BPS - 3rd Ed. Chapter 10 10
Simulation: 1000 sample means Example 10.4
BPS - 3rd Ed. Chapter 10 11
Mean and Standard Deviation of “x-bar”
BPS - 3rd Ed. Chapter 10 12
Mean and Standard Deviation of Sample Means
Since the mean of is , we say that is
an unbiased estimator of X X
Individual observations have standard deviation , but sample means from samples of size n have standard deviation
. Averages are less variable than individual observations.
X
n
BPS - 3rd Ed. Chapter 10 13
Case StudyDoes This Wine Smell Bad?
(Population distribution)
BPS - 3rd Ed. Chapter 10 14
Illustration Exercise 10.8 (p. 258)
Suppose blood cholesterol in a population of men is Normal with = 188 and = 41. You select 100 men at random
(a) What is mean and standard deviation of the 100 x-bars from these samples?– x-barx-bar = 188 (same as µ)
– sx-bar = 41 / sqrt(100) = 4.1 (one-tenth of )
(b) What is probability a given x-bar is less than 180?
Pr(x-bar < 180) standardize z = (180 – 188) / 4.1 = -1.95
= Pr(Z < –1.95) TABLE A = .0256
BPS - 3rd Ed. Chapter 10 15
Central Limit Theorem
No matter what the shape of the population, the sampling distribution of xbar, will follow a Normal distribution when the
sample size is large.
BPS - 3rd Ed. Chapter 10 16
Central Limit Theorem in Action Example 10.6 (p. 259)
Data = time to perform an activity
– NOT Normal (Fig a)– µ = 1 hour – σ = 1 hour
Fig (a) is for single observations
Fig (b) is for x-bars based on n = 2
Fig (c) is for x-bars based on n = 10
Fig (d) is for x-bars based on n = 25
Notice how distribution becomes increasingly Normal as n increases!
BPS - 3rd Ed. Chapter 10 17
Example 10.7 (cont.) Sample n = 70 activities What is the distribution of x-bars?
– x-barx-bar = 1 (same as µ)– sx-bar = 1 / sqrt(70) = 0.12 (by formula)– Normal (central limit theorem)
What % of x-bars will be less than 0.83 hours?
Pr(x-bar < 0.83) standardize z = (0.83 – 1) / 0.12 = -1.42
Pr(Z < - 1.42) TABLE A = 0.0778
Notice that if Pr(x-bar < 0.83 = 0.0778, then Pr(Z > 0.83) = 1 – 0.0778 = 0.9222
BPS - 3rd Ed. Chapter 10 18
BPS - 3rd Ed. Chapter 10 19
Statistical process control*
Skip pp. 262 – 269