January 14, 2002

Statistical Inference

 

Definitions

 

  1. The design of a sample refers to the method that we use to choose the sample from the population.
  2. A simple random sample (SRS) of size n consists of n individuals or objects from the population chosen in such away that every set of n individuals has the equal chance to be selected.
  3. The random variables form a simple random sample (SRS), or are independent and identically distributed (iid), if
  1. The Xi’s are independent random variables.

II.   Every Xi has the same probability distribution.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Recall the following 2 theorems from the Probability and Statistics I class:

 

The Distribution of the Sample Mean

 

Theorem. If X1,…,Xn is a SRS from a normal distribution with mean m and standard deviation s, then

Theorem (The Central Limit Theorem, CLT) If X1,…,Xn is a SRS from a distribution with mean m and standard deviation s, then, if n is sufficiently large (n>30),

 

Exercise 1. The foot is a length measure that was originally introduced as a length of an average human foot. Since the length of a foot differs among people, Germans in sixteenth century calculated the average length of a foot for sixteen men selected at random, and this average became the standard for “the correct foot”. Length of a foot of a man is a random variable with mean 262.5 mm and standard deviation 12 mm.

A.     If the length of a foot is assumed to have a normal distribution, what is the probability distribution of the average foot length of 16 randomly selected men? (hint: the average foot length of 16 men, )

B.     Find the probability distribution that the average foot length of 100 randomly selected men. Do we have to make an assumption about normality of the population distribution, as in part A, to find the distribution?

C.     Find the probability that the average foot length of 100 randomly selected men exceeds 264 mm.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Point Estimation.

I. Bias and Variability

 

Distribution of the sample mean for n=20 and n=100

Distribution of the sample mean, Xi~N(5,1), n=20, number of repetitions = 1,000

Distribution of the sample mean, Xi~N(5,1), n=100, number of repetitions = 1,000

Distribution of estimators of the variance

Distribution of the sample variance, Xi~N(100,50), n=10, number of repetitions = 1,000; mean of sample variances: 2504.052

Distribution of the sample variance*(n-1)/n,, Xi~N(100,50), n=10, number of repetitions = 1,000;

Mean of sample variances: 2238.597

Distribution of estimators of the parameter q for the uniform [0, q] distribution.

2*sample mean, Xi~U(0,5), number of repetitions = 1,000; mean: 5.002624

((n+1)/n)*max(xi), Xi~U(0,5), number of repetitions = 1,000; mean: 5.000522