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

2. Examples Population Samples
Students pursuing undergraduate engineering degrees
Cars capable of speeds in excess of 160 mph.
Potato chips produced at the Frito-Lay plant in Kathleen
Freshwater lakes and rivers
Samples
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.
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

3. Basic Statistics (review)
1. Sample Mean:
Example:
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:
= ___________________
XQ = 87.5
XS = 85 Q S 92 85 95 88 75 78
EGR Ch. 8

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

5. Your Turn
Work in groups of 4 or 5. Find the mean, variance, and standard deviation for your group of the (approximate) number of hours spent working on homework each week.
EGR Ch. 8

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

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

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

9. Example:
The time to respond to a request for information from a customer help line is uniformly distributed between 0 and 2 minutes. In one month 48 requests are randomly sampled and the response time is recorded.
What is the probability that the average response time is between 0.9 and 1.1 minutes?
μ =______________ σ2 = ________________
μX =__________ σX2 = ________________
Z1 = _____________ Z2 = _______________
P(0.9 < X < 1.1) = _____________________________
f(x) = ½, 0<x<2 (uniform dist)
μ = (b-a)/2 = 1 σ2 = (b-a)2/12 = 1/3
z = (x-μ)
(σ/sqrt(n))
μx = 1 σx2 = (1/3)/48 = 1/144
Z1 = (.9-1)/(1/12) = -1.2
Z2 = (1.1-1)/(1/12) = 1.2
P(Z2) – P(Z1) = =0.7698
EGR Ch. 8

10. 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

11. Sampling Distribution of S2
Given:
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

12. 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

13. Example Look at example 8.10, pg. 218: μ = 3 σ = 1 n = 5
s2 = ________________
χ2 = __________________
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
EGR Ch. 8

14. 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 =
EGR Ch. 8

15. 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

16. 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

17. 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

18. Using the t-Distribution
Testing assumptions about the value of μ
Example: problem 14, pg. 228
What value of t corresponds to P(t < tα) = 0.95?
x-bar = 0.475
s2 = sum(x – x-bar)2/(n-1) =
s =
μ= 0.5
t = ( )/(0.1832/sqrt(8)) = -0.39
Looking in table a.4, pg. 672 for an α value associated with (n-1) = 7 degrees of freedom and a t-value of 0.39 (by symmetry), we see it is somewhere between 0.4 and 0.3  call it 0.35.
P(xbar < 0.5) = P(T<-.39) ≈ 0.35 … inconclusive
EGR Ch. 8

19. 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

20. 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

21. Example Problem 17, pg. 228 S12 = ___________________
F = _____________ f0.05 (4, 5) = _________
NOTE:
Note: if the two population variances are equal, then F=S12 / S22 so we are testing the hypothesis that F=S12 / S22 = 1
S12 =(5*SUMSQ(A1:A5)-SUM(A1:A5)^2)/(5*4) = 15750
S22 =(6*SUMSQ(B1:B6)-SUM(B1:B6)^2)/(5*6) = 10920
F = 15750/10920 =1.44
f0.05 (4, 5) = 5.19
EGR Ch. 8