1. 5.5.3 Means and Variances for Linear Combinations

2. What is called a linear combination of random variables?
aX+b, aX+bY+c, etc.
2X X+9Y+10

3. If a and b are constants, then
E(aX+b)=aE(X)+b.
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Corollary 1. E(b)=b.
Corollary 2. E(aX)=aE(X).

10. 5.5.5 The Central Limit Effect
If X1, X2,…, Xn are independent random variables with mean m and variance s2, then for large n, the variable is approximately normally distributed.

11. If a population has the N(m,s) distribution, then the sample mean x of n independent observations has the distribution.
For ANY population with mean m and standard deviation s the sample mean of a random sample of size n is approximately
when n is LARGE.

12. Central Limit Theorem
If the population is normal or sample size is large, mean follows a normal distribution
and
follows a standard normal distribution.

13. The closer x’s distribution is to a normal distribution, the smaller n can be and have the sample mean nearly normal.
If x is normal, then the sample mean is normal for any n, even n=1.

14. Usually n=30 is big enough so that the sample mean is approximately normal unless the distribution of x is very asymmetrical.
If x is not normal, there are often better procedures than using a normal approximation, but we won’t study those options.

16. Even if the underlying distribution is not normal, probabilities involving means can be approximated with the normal table.
HOWEVER, if transformation, e.g. , makes the distribution more normal or if another distribution, e.g. Weibull, fits better, then relying on the Central Limit Theorem, a normal approximation is less precise. We will do a better job if we use the more precise method.

18. Example X=ball bearing diameter
X is normal distributed with m=1.0cm and s=0.02cm
=mean diameter of n=25
Find out what is the probability that will be off by less than 0.01 from the true population mean.

19. Exercise
The mean of a random sample of size n=100 is going to be used to estimate the mean daily milk production of a very large herd of dairy cows. Given that the standard deviation of the population to be sampled is s=3.6 quarts, what can we assert about he probabilities that the error of this estimate will be more then 0.72 quart?