University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2024
Views:

224

Pages:

11
Author:

Gilbert Woodward

Basic Probability Distributions Part 2 How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality Albert Einstein Point Estimation  x as Point Estimator of µ = x   x ∑= i 30 32,910 = 1097 30 s as Point Estimator of σ 2 ( x − x ) ∑= i = s 29 p as Point Estimator of p 163, 996 = 75.2 29 = p 20 = 30 .67 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Point Estimation Once all the data for the 900 applicants were entered in the college’s database, the values of the population parameters of interest were calculated.  Population Mean SAT Score xi ∑ = µ = 1090 900  Population Standard Deviation for SAT Score  2 ( x µ ) − ∑ i σ = 80 = 900 Population Proportion Wanting On-Campus Housing 648 = p = .72 900 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value µ = Population mean 1090 SAT score Point Estimator Point Estimate x = Sample mean 1097 SAT score σ = Population std. 80 s = Sample standard deviation for SAT score 75.2 p = Population proportion wanting campus housing .72 p = Sample proportion wanting campus housing .67 deviation for SAT score Practical Advice The target population is the population we want to make inferences about. The sampled population is the population from which the sample is actually taken. Whenever a sample is used to make inferences about a population, we should make sure that the targeted population and the sampled population are in close agreement. Sampling Distribution of x  Process of Statistical Inference Population with mean µ=? The value of x is used to make inferences about the value of µ. A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean x. Sampling Distribution of x The sampling distribution of x is the probability distribution of all possible values of the sample mean x . • Expected Value of x E( x ) = µ where: µ = the population mean When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased. Sampling Distribution of x • Standard Deviation of x We will use the following notation to define the standard deviation of the sampling distribution of x . σx = the standard deviation of x σ = the standard deviation of the population n = the sample size N = the population size Sampling Distribution of x • Standard Deviation of x Finite Population N −n σ σx = ( ) N −1 n Infinite Population σx = σ n • A finite population is treated as being infinite if n/N < .05. • ( N − n) / ( N − 1) is the finite population correction factor. • σ x is referred to as the standard error of the mean. Sampling Distribution of x When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size. In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size 30 or more. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed. Sampling Distribution of x The sampling distribution of can be used to provide probability information about how close the sample mean is to the population mean µ .

Basic Probability Distributions, Part 2

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock