1. Populations and Samples
Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement”
Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole”
(Merriam-Webster Online Dictionary, October 5, 2004)
EGR Ch. 8 8th edition
Spring 2008

2. Examples Population Samples
Students pursuing undergraduate engineering degrees
Cars capable of speeds in excess of 160 mph.
Samples
1000 engineering students selected at random from all engineering programs in the US
50 cars selected at random from among those certified as having achieved 160 mph or more during 2003
Samples:
10 chips selected at random every 5 minutes as the conveyor passes the inspector.
4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers
OTHERS?
EGR Ch. 8 8th edition
Spring 2008

3. Examples (cont.) Population Samples
Potato chips produced at the Frito-Lay plant in Kathleen
Freshwater lakes and rivers
Samples
10 chips selected at random every 5 minutes as the conveyor passes the inspector
4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers
Samples: engineering students selected at random from all engineering programs in the US.
50 cars selected at random from among those certified as having achieved 160 mph or more during 2003.
OTHERS?
EGR Ch. 8 8th edition
Spring 2008

4. Basic Statistics (review)
Sample Mean:
At the end of a team project, team members were asked to give themselves and each other a grade on their contribution to the group. The results for two team members were as follows:
= ___________________ 87.5
= ___________________ 85.0 Q S 92 85 95 88 75 78
XQ = 87.5
XS = 85
EGR Ch. 8 8th edition
Spring 2008

5. Basic Statistics (review)
1. Sample Variance:
For our example:
SQ2 = ___________________
SS2 = ___________________
S2Q =
S2S = Q S 92 85 95 88 75 78
S2Q =
S2S =
EGR Ch. 8 8th edition
Spring 2008

6. Sampling Distributions
If we conduct the same experiment several times with the same sample size, the probability distribution of the resulting statistic is called a sampling distribution
Sampling distribution of the mean: if n observations are taken from a normal population with mean μ and variance σ2, then:
EGR Ch. 8 8th edition
Spring 2008

7. Central Limit Theorem Given: Then,
X : the mean of a random sample of size n taken from a population with mean μ and finite variance σ2,
Then,
the limiting form of the distribution of
is the standard normal distribution n(z;0,1)
EGR Ch. 8 8th edition
Spring 2008

8. Central Limit Theorem
If the population is known to be normal, the sampling distribution of X will follow a normal distribution.
Even when the distribution of the population is not normal, the sampling distribution of X is normal when n is large.
NOTE: when n is not large, we cannot assume the distribution of X is normal.
EGR Ch. 8 8th edition
Spring 2008

9. Sampling Distribution of the Difference Between Two Averages
Given:
Two samples of size n1 and n2 are taken from two populations with means μ1 and μ2 and variances σ12 and σ22
Then,
See example 8.8, pg 213 and example 8.9, pg 214
EGR Ch. 8 8th edition
Spring 2008

10. Sampling Distribution of S2
Given:
If S2 is the variance of of a random sample of size n taken from a population with mean μ and finite variance σ2,
Then,
has a χ2 distribution with ν = n - 1
EGR Ch. 8 8th edition
Spring 2008

11. χ2 Distribution
χα2 represents the χ2 value above which we find an area of α, that is, for which P(χ2 > χα2 ) = α.
EGR Ch. 8 8th edition
Spring 2008

12. Example
Look at example 8.10, pg. 256: A manufacturer of car batteries guarantees that his batteries will last, on average, 3 years with a standard deviation of 1 year. A sample of five of the batteries yielded a sample variance of Does the manufacturer have reason to suspect the standard deviation is no longer 1 year?
μ = 3 σ = 1 n = 5 s2 = 0.815
If the χ2 value fits within an interval that covers 95% of the χ2 values with 4 degrees of freedom, then the estimate for σ is reasonable.
(See Table A.5, pp )
from book (& excel) – s2 = 0.815
χ2 = (n-1)s2 / σ2
= (4)(0.815)/1 = 3.26
looking for X2 values that cover 95%, meaning α values between and 0.975
from table A.5 => Χ =11.143
Χ = 0.484
Χ = Χ = 0.484
EGR Ch. 8 8th edition
Spring 2008

13. Your turn …
If a sample of size 7 is taken from a normal population (i.e., n = 7), what value of χ2 corresponds to P(χ2 < χα2) = 0.95? (Hint: first determine α.)
NOTE the figure associated with table A.5!! (These values cover areas > the X2 value …)
ν= 7-1 = 6; α = 0.05
X2 =
12.592
EGR Ch. 8 8th edition
Spring 2008

14. t- Distribution Recall, by CLT: is n(z; 0,1)
Assumption: _____________________
(Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …)
assumption: we know σ
EGR Ch. 8 8th edition
Spring 2008

15. What if we don’t know σ? New statistic: Where, and
follows a t-distribution with ν = n – 1 degrees of freedom.
EGR Ch. 8 8th edition
Spring 2008

16. Characteristics of the t-Distribution
Look at fig. 8.11, pg. 221***
Note:
Shape: _________________________
Effect of ν: __________________________
See table A.4, pp
shape – symmetrical about 0
effect of ν – variance, as seen in the width of the curve, depends on sample size
note – as ν increases, curve looks more like a normal distribution (hence, the CLT)
Table A.4 – critical values of t for several values of α and df. Note that the table yields the right tail of the distribution.
EGR Ch. 8 8th edition
Spring 2008

17. Comparing Variances of 2 Samples
Given two samples of size n1 and n2, with sample means X1 and X2, and variances, s12 and s22 …
Are the differences we see in the means due to the means or due to the variances (that is, are the differences due to real differences between the samples or variability within each samples)?
See figure 8.16, pg. 226
EGR Ch. 8 8th edition
Spring 2008

18. F-Distribution Given: Then,
S12 and S22, the variances of independent random samples of size n1 and n2 taken from normal populations with variances σ12 and σ22, respectively,
Then,
has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom.
(See table A.6, pp )
EGR Ch. 8 8th edition
Spring 2008