7-1 and 7-2 Sampling Distribution Central Limit Theorem

Let’s construct a sampling distribution (with replacement) of size 2 from the sample set {1, 2, 3, 4, 5, 6} 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

Mean =  Probability 11/ /36 23/ /36 35/ /36 45/ /36 63/ /36 61/36

Theorem 7-1 Some variable x has a normal distribution with mean = μ and standard deviation = σ For a corresponding random sample of size n from the x distribution - the  distribution will be normal, - the mean of the  distribution is μ - the standard deviation is

What does this mean? If you have a population and have the luxury of measuring a lot of sample means, those means (called xbar) will have a normal distribution and those means have a mean (i.e. average value) of mu.

For the sample size 2 What is the mean of {1, 2, 3, 4, 5, 6}? What appear to be the mean of the distribution of 2 out of 6?

Theorem 7-1 (Formula) Why doesn’t the SD stay the same? Because the sample size is smaller… you will see a smaller deviation than you would expect for the whole population

Central Limit Theory Allows us to deal with not knowing about original x distribution (Central = fundamental) The Mean of a random sample has a sampling distribution whose shape can be approximated y the Normal Model as the value of n increases. Larger Sample = Bigger Approximation The standard is that n ≥ 30.

Example Coal is carried from a mine in West Virginia to a power plant in NY in hopper cars on a long train. The automatic hoper car loader is set to put 75 tons in each car. The actual weights of coal loaded into each car are normally distributed with μ = 75 tons and σ = 0.8 tons.

What is the probability that one car chosen at random will have less than 74.5 tons of coal? This is a basic probability – last chapter

What is the probability that 20 cars chosen at random will have a mean load weight of less than 74.5 tons? The question here is that the sample of 20 cars will have  (xbar) ≤ 74.5

Another Example Invesco High Yield is a mutual fund that specializes in high yield bonds. It has approximately 80 or more bonds at the B or below rating (junk bonds). Let x be a random variable that represents the annual percentage return for the Invesco High Yield Fund. Based on information, x has a mean μ = 10.8% and σ = 4.9%

Why would it be reasonable to assume that x (the average annual return of all bonds in the fund) has a distribution that is approximately normal? 80 is large enough for the Central Limit Theorem to apply

Compute the probability that after 5 years  is less than 6% (Would that seem to indicate that μ is less than 10.8% and that the junk bond market is not strong?) N = 5 because we are looking over 5 years Yes. The probability that it is less than 6% is approx. 1%. If it is actually returning only 6%, then it looks like the market is weak.

Compute the probability that after 5 years  is greater than 16%

Note The Normal model applies to quantitative data…