MORE PROBABILITY AND SAMPLE PROPORTIONS

Probability

Talking about chance. What are the chances of rain tomorrow? What are the chances of winning the California Lottery? Probability theory was developed to answer these questions (in fact, most of the early work was done by gamblers...)

The basic idea is simple, if you have some kind of random phenomenon -- a coin toss -- you might not be able to say how the next toss will fall, but you might be able to say with some certainty how many heads you will get if you toss a coin 20 times (about 10 heads).

Definition of Random from the text: Individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.

Definition of Probability: given some random phenomenon this is the proportion of times the outcome would occur over many repetitions.

In practice, we aren't able to draw many many samples or repeat some experiment many times. Instead, we rely on probability theory.

a. A probability is a number between 0 and 1

b. All possible outcomes must have probability 1

c. The probability that an event does not occur is 1 minus the probability that it does occur.

d. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

Try question 4.13

Counts and Proportions

Proportions generally involve "yes/no" outcomes. From your book "Do you find clothes shopping frustrating and time-consuming?" This is something retailers want to know -- what proportion or what percentage -- find clothes shopping frustrating? What they want to know is the population parameter p.

If they draw a random sample of 2500 and ask...and suppose 1650 find shopping frustrating. The estimate, the statistic, is p-hat or .66

Sampling Distribution of a Sample Proportion

If n is large, the distribution of the sample proportion will be approximately normal with mean = p and standard deviation sigma = sqrt[p(1-p)/n].

Remember, this formula for the standard deviation work when the population is at least 10 times larger than the sample.

The Sample Proportion and Calculations using the Standard Normal Curve

Now that you have the sample proportion and its standard deviation and n is large. What now? One can start to talk about how close the outcome is to the "truth".

See example 4.15 in your book (pp 272-273)

Sample Counts

Sometimes a question is expressed in terms of counts or sometimes you are interested in an exact count. To work with these situations, restate them in terms of proportions.

Example from problem 4.49