5.5.3 Means and Variances for Linear Combinations

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Presentation transcript:

5.5.3 Means and Variances for Linear Combinations

What is called a linear combination of random variables? aX+b, aX+bY+c, etc. 2X+3 5X+9Y+10

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

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.

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.

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

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.

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.

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.

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.

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?