BPS - 3rd Ed. Chapter 101 Sampling Distributions

BPS - 3rd Ed. Chapter 10 1

Chapter 10

Sampling Distributions


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


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!


The Law of Large Numbers


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?


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


Simulation: law of large numbers Fig 10.1

The sample mean gets close to population mean as we take more and more samples


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?


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?


Simulation: 1000 sample means Example 10.4


Mean and Standard Deviation of “x-bar”


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


Case StudyDoes This Wine Smell Bad?

(Population distribution)


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


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.


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!


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



Statistical process control*

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BPS - 3rd Ed. Chapter 101 Sampling Distributions