{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ch5 - STAT 200 Chapter 5 Sampling Distributions The...

This preview shows pages 1–3. Sign up to view the full content.

STAT 200 Chapter 5 Sampling Distributions The binomial distribution for counts (Section 5.1) 1. The Binomial Experiment (a) The experiment consists of n identical trials. The number of trials is fixed in advance. (b) The outcomes on each trial can be dichotomized into two complementary cate- gories: “success” and “failure”. (c) The probability of the success event, p , is the same from trial to trial. The probability of the failure event is 1 - p . (d) The n trials are independent of each other. In other words, the results of previous trials do not affect the result of the current trial. Under the conditions of a binomial experiment, the random variable X representing the number/count of successes out of the n trials is a binomial random variable with parameters n and p . We write X Bin ( n, p ). 2. For X Bin ( n, p ), the probability that X will take on a value of k is given by P ( X = k ) = n k p k (1 - p ) n - k , k = 0 , 1 , 2 , · · · , n where n k = n ! k !( n - k )! is called the binomial coefficient (Another notation is n C k ). It gives the total number of ways (or combinations of successes and failures) of having X = k . Note that k ! = k × ( k - 1) × ( k - 2) × ... × 2 × 1 and 1! = 1, 0! = 1. 3. Mean (expected value), Variance and SD for a Binomial Random Variable X Mean μ X : E ( X ) = n k =0 k × P ( X = k ) = n k =0 k × n k p k (1 - p ) n - k = np 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Variance σ 2 X : V ( X ) = n k =0 ( k - np ) 2 × P ( X = k ) = n k =0 ( k - np ) 2 × n k p k (1 - p ) n - k = np (1 - p ) Standard deviation σ X = SD ( X ) = np (1 - p ) Interpretation of E ( X ): the average number of successes if you are to repeat the binomial experiment (each with n trials) many times. Interpretation of V ( X ): a measure of the variability of the numbers of successes if the binomial experiment (each with n trials) is repeated many times. Sampling distribution (Section 3.3) Population vs. sample, parameter vs. statistic The complete collection of individuals that we want to study about is called the population . A census provides a means to obtain complete and accurate informa- tion about a population of interest. However, it can be very costly and time inefficient to carry out. Sometimes a census is impossible. It is more practical to study a sample , i.e., a subset of individuals selected from a population. A sample can provide reliable information about a population as long as the sample is representative of the population. A systematically nonrepresentative sample is said to be biased . By randomly selecting individuals from the population, the sample produced tend to have characteristics comparable to the population. The chance of obtaining a non- representative sample is thus minimized. A parameter refers to a numerical summary of a population. The counterpart of a sample is a statistic . The value of a parameter is fixed yet unknown in practice. We use statistics to estimate population parameters. Due to sampling variability (varia- tion from sample to sample), a statistic takes on different values for different samples.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 11

ch5 - STAT 200 Chapter 5 Sampling Distributions The...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online