1. The Behavior of Sums and Averages Amore Frozen Foods
Class 12
The Behavior of Sums and Averages
Amore Frozen Foods
EMBS (selections from)

2. Review Class 11. chi-squared test of independence
H0: The Categorical variables are independent
Ha: They are not.
Pivot Table to create contingency table.
Excel to create Expected Counts
CHITEST() to calculate p-value.
or Excel to get distances and calculated chi-squared statistic, =chidist with dof = (#rows-1)(#col-1) to get p-value

3. There is often another way
H0: Athlete and HSStat are independent
H0: P1=P2
P1 is probability an athlete took HSStat
P2 is the probability a non-athlete took HSStat
You can use the method of section 11.1 to test H0. (check out equation 11.7 to see what’s involved)
And you get exactly the same p-value!!
So, the chi-squared test of independence tests the equality of two probabilities (for a 2x2) and more! That’s why we learned it.

4. Amore
At a target of 8.22, Amore’s filling machine fills mac and cheese tins with an amount N(8.22,0.22)
Resulting in about 16% of tins (pies) being underweight
The FDA requires Amore to test each 20-minute batch using a random sample of five tins. The entire batch must be rejected if the 5-pie sample averages less than 8 ounces.
What is the probability a batch will be rejected?

5. We know how many sample pies will be underfilled….
Binomial(n=5,p=.16) assuming pie weights are independent.
E(number underfilled) = n*p = 5*.16 = 0.8
Var(number underfilled) = n*p*(1-p) = 0.672
Std dev (number underfilled) = 0.820
n is too small to use the normal approximation

6. But the fate of the batch rests on the total (average) weight of the five sampled pies.
We need to figure out what the probability distribution of either the total or the average of five is.

7. For Amore…. 𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + 𝑋 4 + 𝑋 5
We can work our problem either way.
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + 𝑋 4 + 𝑋 5
The sample mean is just the total times 0.2.
𝑋 = 𝑋 1 + 𝑋 2 + 𝑋 3 + 𝑋 4 + 𝑋 5 5
“X-bar”
The sample mean

8. The X’s are independent and identically distributed N(10. 2,
The X’s are independent and identically distributed N(10.2,.22) random variables
8.22,
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + 𝑋 4 + 𝑋 5
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of (Total) =
E(Total) =
Var(Total) =
Standard Deviation (Total) =

9. The X’s are independent and identically distributed N(μ,σ) random variables
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + …+ 𝑋 𝑛
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of (Total) =
Normal
E(Total) =
n*μ
Var(Total) =
n*σ2
Standard Deviation (Total) =
𝑛 𝜎
P(reject Batch) = P(Total<40) = NORMDIST(40,5*8.22,5^.5*.22,true) = 0.013

10. P(reject Batch) = P( 𝑋 <8) = NORMDIST(8,8.22,.22/5^.5,true) = 0.013
The X’s are independent and identically distributed N(μ,σ) random variables
𝑋 = 𝑋 1 + 𝑋 2 + 𝑋 3 + … + 𝑋 𝑛 𝑛
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of ( 𝑋 ) =
Normal
E( 𝑋 ) = μ
Var( 𝑋 ) =
σ2/n
Standard Deviation ( 𝑋 ) =
𝜎/ 𝑛
P(reject Batch) = P( 𝑋 <8) = NORMDIST(8,8.22,.22/5^.5,true) = 0.013

11. The standard deviation of a sample mean ( 𝑋 )
Aka “the standard error of the sample mean”
𝜎 𝑋 = 𝜎 𝑛
(7.3)
A standard ERROR is some other standard deviation divided by 𝑛 somehow.
The formulas hold for n=1 and n=∞ and n’s in between.
𝑋 𝑛 ~𝑁(𝜇, 𝜎 𝑛 )

12. The distribution of a sample mean
Assumes independent, identically distributed Normal random variables.
𝑋 𝑛 ~𝑁(𝜇, 𝜎 𝑛 )

13. Illustration. The lorex fill test data

14. Illustration. The lorex fill test data=

15. Small Schools are Better?
Did Bill Gates waste a billion dollars because he failed to understand the formula for the standard deviation of the sample mean?
The Gates Foundation certainly spent a lot of money, along with many others, pushing for smaller schools and a lot of the push came because people jumped to the wrong conclusion when they discovered that the smallest schools were consistently among the best performing schools.

16. Small School dominate the “top 25”

17. ….and the bottom!

18. He should have taken “Statistics in Business”
In recent years Bill Gates and the Gates Foundation have acknowledged that their earlier emphasis on small schools was misplaced.
He should have taken “Statistics in Business”

19. The X’s are independent and identically distributed N(μ,σ) random variables
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + …+ 𝑋 𝑛
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of (Total) =
Normal
E(Total) =
n*μ
Var(Total) =
n*σ2
Standard Deviation (Total) =
𝑛 𝜎

20. The X’s are independent and identically distributed N(μ,σ) random variables
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + …+ 𝑋 𝑛
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of (Total) =
E(Total) =
Var(Total) =
Standard Deviation (Total) =

21. The X’s are independent and identically distributed N(μ,σ) random variables
𝑇𝑜𝑡𝑎𝑙= 𝑋 1 + 𝑋 2 + 𝑋 3 + …+ 𝑋 𝑛
Means and Variances nicely Add. Adding standard deviations is very bad.
The Distribution of (Total) = ??
E(Total) =
μ1 + μ2 + … μn
Var(Total) =
σ21+ σ22+…. σ2n
Standard Deviation (Total) =
𝑉𝑎𝑟

22. Illustration: Suppose Wunderdog did five sports
Number correct if guessing
Expected # correct
Variance
Std Dev 1
149 X1
74.5
37.25
6.10 2 85 X2
42.5
21.25
4.61 3
133 X3
66.5
33.25
5.77 4
209 X4
104.5
52.25
7.23 5
311 X5
155.5
77.75
8.82
Total
887
sum of the Xs
443.5
221.75
14.89
Means and Variances nicely add. Adding standard deviations is very bad.

23. Illustration: Suppose Wunderdog did five sports
Number correct if guessing
Expected # correct
Variance
Std Dev 1
149 X1
74.5
37.25
6.10 2 85 X2
42.5
21.25
4.61 3
133 X3
66.5
33.25
5.77 4
209 X4
104.5
52.25
7.23 5
311 X5
155.5
77.75
8.82
Total
887
sum of the Xs
443.5
221.75
14.89
Means and Variances nicely add. Adding standard deviations is very bad.
=[887*.5*.5]^.5
=887*.5
=887*.5*.5
The Binomial Distribution “knows” that means and variances add.

24. Example Problem EMBS 40 p 306.
A large BW survey of MBA’s ten years out found mean dining out spending to be $115.5 (per week) and population standard deviation of $35. You will do a follow-up study and survey a random sample of 40 of the MBA alums.
What sample mean will you get?
What is the probability your sample mean will be within $10 of the population mean?
What is the probability your sample mean will be less than $100?

25. Summary
If we know the X’s are independent, identically distributed N(μ,σ) random variables…
The total of n X’s will be N(nμ, 𝑛 σ)
The sample mean of n X’s is written as
will be N(μ, σ/ 𝑛 )
Means and variances nicely add (if the X’s are independent). Adding standard deviations is very bad.
The BINOMIAL knows this…..
𝑋 𝑛
𝑋 𝑛

26. Exam1 Complete answers have been posted
The Binomial works for small n.
P(Cubs Win) = BINOMDIST(3,3,.5,false)
P(2Gin4) = BINOMIDIST(2,4,p,false)
Whether p is 0.5 or 0.48
P(2P l noD) = BINOMDIST(2,2,.2,false)
P(1P l noD) = BINOMDIST(1,2,.2,false)
P(0P l noD) = BINOMDIST(0,2,.2,false)
#correctly filled is B(n=144,p=.92775)