Introduction The Sample Mean The Central Limit Theorem
The Sample Variance Sampling Distribution from The Normal Distribution Sampling from A Finite Population
◦ Distribution of The Sample Mean ◦ Joint Distribution of X and S2
◦ Approximate Distribution of The Sample Mean ◦ How Large A Sample Is Needed
Recall definitions of:
◦ ◦ ◦ ◦ ◦ ◦ ◦ Population Sample Inferential statistics Sampling Random sampling Parameter Statistic
If X1, . . . , Xn are independent random variables having a common distribution F, g , then we say that they constitute a sample (sometimes called a random sample) from the distribution F.
Parametric inference problem: Problems in which the form of the underlying distribution is specified up to a set of unknown parameters, eg: Nonparametric inference problem: problems in hi h i which nothing is assumed about the form thi i d b t th f of ...view middle of the document...
Then for n large, the distribution of
X1 +・ ・ ・+Xn is approximately normal with mean nμ and
variance nσ2. i
The Central Limit Theorem
Problem: An insurance company has 25,000 automobile policy holders If the yearly claim holders. of a policy holder is a random variable with mean 320 and standard deviation 540, approximate the probability that the total yearly claim exceeds 8.3 million.
Since the sample mean has expected value μ and standard deviation σ/√n it then follows √n, that
has approximately a standard normal distribution
The weights of a population of workers have mean 167 and standard deviation 27. (a) If a sample of 36 workers is chosen, approximate the probability that the sample mean of their weights lies between 163 and 171. (b) Repeat part (a) when the sample is of size 144. 144
A general rule of thumb is that one can be confident of the normal approximation whenever the sample size n is at least 30 30. That is, practically speaking, no matter how non normal the underlying population distribution is, the sample mean of a sample of size at least 30 will be approximately normal. normal In most cases, the normal approximation is valid for much smaller sample sizes
Theorem If X1, . . . , Xn is a sample from a normal s sa p e o o a population having mean μ and variance σ2,
◦ X and S2 are independent random variables, with X being normal with mean μ and variance σ2/n and (n − 1)S2/σ2 being chi square with n − 1 degrees of chi-square freedom.
The time it takes a central processing unit to process a certain type of job is p yp j normally distributed with mean 20 seconds and standard deviation 3 seconds. If a sample of 15 such jobs is observed, what is the probability that the sample variance will exceed 12?
A sample of size n from this population is said to be a random sample if it is chosen in such a manner that each of the C(N, n) population subsets of size n is equally likely to be the sample.