Math 3680

Lecture #11

The Central Limit Theoremfor Sample Counts and

Sample Proportions

The Box Model

Definition.

In a box model, the population is represented by the contents of the box.

A sampling experiment is the draw of tickets from the box.

The sample consists of the tickets that are drawn.

The Box Model

Big Idea:

Find an analogy between the process being studied and drawing numbers from a box.

 

Definition. A simple random sample consists of n observations X1, X2, …, Xn which are drawn sequentially, either with or without replacement .

A function of a random sample is called a statistic. A statistic is itself a random variable. Common examples of statistics are:

1

)()()( 222

212

21

21

n

XXXXXXS

n

XXXX

XXXSUM

n

n

n

This way of thinking about statistics can be a stretch of the imagination. Ordinarily, you think of statistics in such statements as “the sum of 25 dice is 80,” or “the average of 25 dice is 3.2.”

Instead, you should think about a statistic, such as the sum of 25 dice, as a random quantity which can change from sample to sample: “the sum of 25 dice is a number between 25 and 150 with a certain distribution.”

Do not think of a statistic as the specific outcome of a specific simple random sample.

Just like other random variables, we can compute the mean and standard deviation of statistics.

We’ll discuss these calculations in more detail when we discuss the Law of Averages and the Central Limit Theorem.

Example: Thirty draws are made at random from a box that has two tickets; one labeled “1”, and the other labeled “2”. (A) How small can the sum be? How large can the sum be?

(B) About how many times do you expect the “1” ticket to appear? The “2” ticket?

(C) About how much do you expect the sum to be?

(D) Is there a difference if you draw from a box that contains 4 tickets: two labeled “1”, and two labeled “2”?

How to Make a Box Model

1. What numbers go into the box?

2. How many of each?

3. How many draws?

Example: Suppose a gambler gambles on a roulette table 10 times, betting $1 on black each time. The roulette wheel has 38 places: 18 black, 18 red, and 2 green. Construct the box model.

Example: You bet 100 times on a single number, say 19. Betting on a single number pays 35 to 1. In other words, if you bet a dollar and win, you get your dollar back and 35 more.

 

Fill in the blanks: The net gain will be like the sum of __________ draws from a box that contains___________________.

Example: You bet 100 times on a single number, say 19. Betting on a single number pays 35 to 1. In other words, if you bet a dollar and win, you get your dollar back and 35 more.

 

Fill in the blanks: The number of wins will be like the sum of __________ draws from a box that contains___________________.

What are some advantages of the box model? It strips the roulette wheel and the table to its bare essentials:

 

1. The tickets: the possible outcomes of each spin.

2. The number of tickets: the probability of each outcome.

3. The number of draws the number of plays.

The sum of the draws provides the net gain.

The Law of Averages

for the Sample Count

Theorem. Suppose random variables X1, X2, …, Xn are drawn with replacement from a box of “0”s and “1”s. Suppose the proportion of “1”s in the box is , so that the mean of the population is . Let

K = X1 + X2 + …+ Xn.

Then E(K) = n and SD(K) = (Why?)

This latter quantity is also called the standard error.

Furthermore, if both n 5 and n (1 - 5, then we may approximate probabilities of K by converting to standard units and using the normal curve and the continuity correction.

.)1( n

Theorem. Suppose random variables X1, X2, …, Xn are drawn without replacement from a box of size N which contains only “0”s and “1”s. Suppose the proportion of “1”s in the box is , so that the mean of the population is . Let

K = X1 + X2 + …+ Xn.

Then E(K) = n and SD(K) =

(Why?)

Furthermore, if both n 5 and n (1 - 5, then we may approximate probabilities of K by converting to standard units and using the normal curve and the continuity correction.

.)1(1

nN

nN

Example: Of all people with a certain nervous disorder, 90% eventually make a full recovery. Seventy-five patients are currently hospitalized. What is the chance that fewer than 65 recover?

Example: Of all people with a certain nervous disorder, 90% eventually make a full recovery. Ten patients are currently hospitalized. What is the chance that 9 or more recover?

Example: A university has 5,000 students, which consists of 45% men and 55% women. A simple random sample of 500 students is selected. What is the chance that fewer than 200 are men?

Example: A university has 50,000 students, which consists of 45% men and 55% women. A simple random sample of 500 students is selected. What is the chance that fewer than 200 are men?

The Law of Averages

for the Sample Proportion

Some questions are posed with regard to the sample proportion (as opposed to the sample count). Such problems can be easily converted into questions regarding the sample count, which can then be analyzed using the techniques we’ve already discussed.

Example: According to the Census, a certain city has a population of 100,000 people age 18 and over. Of these, 60% are married.

A sample of size 1000 is drawn from the population. Find the chance that 58% or less of the people in the sample are married.