Math 37 - Lecture 15 (really this time!)

Properties of (5.1)

Example 1 One test for Extrasensory Perception is for an experimenter to use a set of 5 cards with different symbols. The experimenter selects one card from the deck at random and stares at the symbol without allowing the subject to see the symbol. If the subject specifies the correct symbol, that is a "success". The experimenter records the number of successes in several trials to see if there is convincing evidence of ESP.

a) Why does the experimenter take more than one observation?

b) Is this a categorical or quantitative measurement?

c) Let X=number of correct responses, what are the properties of X?

d) If the subject is just guessing, what is the probability of success?

e) If 20 trials are taken, what's the expected value of X?

f) How many would the subject have to get correct to convince you of ESP?

g) How does , the proportion of successes, behave when the subject is guessing:

Use your random number table to simulate 20 trials for a pure guesser, calculate . Select another row and calculate a second .

Sampling Distribution of :

 

 

 

Where are these values centered?

What shape does this distribution have?

What spread does this distribution have?

Is this what you expected? Why?

Def - Binomial random variable: X=the count of the number of successes in an experiment, is a Binomial random variable if:

Example 2 Forty percent of all airline pilots are over 40 years of age. Out of a random sample of 15 pilots, let the random variable X be the number of pilots who are over 40 years of age.

Example 3 Suppose a salesperson makes sales to 20% of her customers. One day she counted her customers until she made a sale. Let Y be the number of customers until her first sale.

Example 4 A student takes a sample of 10 Reeses' candies and counts how many are orange.

Rule of Thumb: We can consider "sampling without replacement" a binomial setting when the population is at least 10 times the sample.

Properties of a Binomial Random Variable

 

X

Expected Value

np

p

Variance

np(1-p)

p(1-p)/n

Standard Deviation

 

 

Shape

The Normal Approximation to the Binomial

When the sample size n is large, and the population is huge, the sampling distribution of =X/n is approximately Normal with mean p and standard deviation

*When is the sample size big enough?

Rule of Thumb: np >10 and n(1-p) >10

Example Checking for undercoverage

If know 11% of population are black, how would you evaluate a sample that contained only 9.2% blacks? Is there evidence of undercoverage?

Does Normal approximation apply?

What is the mean and standard deviation?

What is the probability of such a result for an SRS?