Ch 10: Sampling Distributions
D: parameter – a number that describes the population. In statistical practice, the value of the parameter is not
known because we can’t examine the entire population.
D: statistic – a number that can be computed from the sample data without making use of any known
parameters. We often use a statistic to approximate an unknown parameter.
Ex 10.1 Current Population Survey mean income, x = $58, 208 statistic? or parameter?
*statistics come from the sample, parameters come from populations
 = mean of a population (parameter), x = mean of the sample = average of the observations (statistic)
Ex 10.2 DMS (dimethyl sulfide) - “off odors” in wine
 = parameter of the population
estimates 
x =
Law of Large Numbers – Draw observations at random from any population with finite mean,  . As # of
28 40 28 33 20 31 29 27 17 21
observations increases, x of observed values gets closer and closer to the mean,  of the population
Ex 10.3 In Ex 10.2,  = 25 See Fig 10.1
x1  28, x2  40, x2nd 
x3rd 
x4th 
x5th 
Sampling distributions – take a large sample of size 10 from the population. Calculate the sample mean, x for
each sample, make a histogram of x . Examine the distribution of the histogram for shape, center, spread, &
outliers.
D: simulation – computer software to imitate chance behavior
D: sampling distribution of a statistic – the distribution of values taken by the statistic in all possible samples of
the same size from the same population See Fig 10.2 1)shape (looks Normal) 2)Center  = 25, x = 24.95
3) Spread standard deviation = 2.217,  = 7
*If x is the mean of the SRS of size n drawn from a large population with mean,  and standard deviation, ,
then the mean of the sampling distribution of x is  and standard deviation is

n
( M both)
*mean of the statistic , x is the same as the mean,  of the population
x is an unbiased estimator of parameter,  “correct on the average” In many samples, averages are less
variable than individual observations
* If individual observations have a N(,) distribution, then the sample mean, x , of n independent

observations has a N(,
) distribution
n
Ex 10.5 DMS “wine smell”  = 25, n = 10,  = 7 standard deviation =
*results of large samples less variable than results of small samples See Fig 10.3
Central Limit Theorem – Draw an SRS of size n from any population with mean,  and finite standard

deviation, . When n is large, the sampling distribution of the sample mean, x is approx. Normal, N(,
)
n
Exponential
distribution
25 obs
2 obs
10 obs
1 obs
Ex 10.6 See Fig 10.4
Ex 10.7  = 1, n = 70,  = 1 1 A/C unit 70 units Find the probability that the average maintenance time
exceeds 50 min. standard dev =
N(
) 50 min =
hr P(X >
)
Prob, P( x > .83) =
A variable that continues to be described by the same distribution when observed over time is in statistical
control. D: control charts – statistical tools that monitor a process & alert us when the process has been
disturbed or out of control. D: process- monitoring conditions – measure a quantitative variable, x that has a
Normal distribution of X as long as the process is in control
OVER
Ex 10.8 computer monitors involving tension of fine wires - proper tension = 275mV ,  = 43 mV There
were 4 monitors each hour See Table 10.1 standard dev =
Using the 68-95-99.7 Rule, 99.7% were between?
x control chart : 1) plot means, x of regular samples of size n against time
2) draw horizontal center line at  3) draw horizontal control limits at   3 / n (Any x that doesn’t fall
between the control limits is “out of control” See Fig 10.6
Ex 10.9 In Fig 10.6, no “out of control” values Now see Fig 10.7 & Fig 10.8 “out of control”
*control charts focus on process itself rather than individual products
D: natural tolerances – The 68-95-99.7 Rule tells us that almost all measurements will lie in the range   3 
Do not mix this up with the range   3 / n for individual measurements of expected range of sample means!
Ex 10.10 In Ex 10.8, find the natural tolerances where proper tension = 275mV ,  = 43 mV