Binomial Distribution and Random Variable



5 - Discrete Random Variables and Probability Distributions

Motivating Example and Definitions:

Binomial Distribution and Random Variable

A binomial random variable, X, is defined to be the number of “successes” in n independent trials where the P(success) = p is constant for each of trials. In the definition above notice the following conditions need to be satisfied for a binomial experiment:

1. There is a fixed number of n trials carried out.

2. The outcome of a given trial is either a “success” or “failure”.

3. The probability of success (p) remains constant from trial to trial.

[pic]= q

4. The trials are independent, that is the outcome of a given trial is not affected by the outcome of any other trial.

Binomial Probability Function

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The factorial expression in front counts the number of ways it is possible to obtain x successes in n trials while the remainder of the formula gives the probability of getting a particular sequence with x “successes” and n – x “failures”.

The JMP spreadsheet Binomial Table Generator does all the probability calculations for the following:

• P(X = x) = probability of obtaining x “success” in n trials.

• P(X < x) = probability of obtaining x or less “successes” in n trials.

• P(X > x) = probability of obtaining x or more “successes” in n trials.

To use the Binomial Table Generator file in JMP you simply need to change the number of independent trials (n) and the probability of success (p) in the table below. To change the n and p values right-click at the top of the column and select Formula. You can then change the number in the formula to your desired values for n and p. The table automatically updates and gives the following probabilities in the last three columns. Be sure to look at the comments on each in the upper left-hand corner of the table.

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Example 1: Drug Side Effects

A drug company claims that 10% of patients taking the drug will experience adverse side effects. To test this claim a researcher administers the drug to a random sample of 20 patients. Let X = the number patients in our sample who experience adverse side effects.

To set up the binomial proability table for this situation we need to change the number of trials to n = 20 and the probability of success to p = .10, to do this right-click on the top of each column and select Formula.

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a) What is the probability that exactly 4 patients experience side effects?

b) What is the probability that 2 patients or less experience side effects?

c) What is the probability that 5 patients or less experience side effects?

d) What is the probability that 6 or more patients experience side effects?

e) Suppose that in our sample 6 patients experience side effects. What do think about the claim made by the drug company on the basis of this result?

f) Suppose we gave the drug to 100 patients. If the drug company’s claim regarding side effects is true, how many side effects do we expect to observe, i.e. what is the mean or expected value of X?

For a binomial random variable [pic]

g) Find the standard deviation of the number of side effects and use it to give an interval that is very likely to cover the number of side effects observed when giving the drug to n = 100 patients.

For a binomial random variable [pic]

When n is sufficiently large then X = # of “successes” in the n trials is approximately normally distributed. As an example consider the histogram below which shows the simulated results of 10,000 clinical trials where for each trial n = 100 patients are given a drug which has [pic] or a 10% chance of causing a side effect and the number of patients experiencing side effects is observed.

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h) How many side effects would you have to observe to be convinced that the 10% side effect rate is wrong and that the true side effect rate is greater?

Binomial Distribution Table for n = 100 and p = .10

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Example 2: Effect of Togetherness on Heart Rates of Rats (SIGN TEST)

A researcher was interested in determining if the heart rate of rats increases when they are in a cage with other rats versus when they are in a cage by themselves. The researcher thought this would be the case but wanted to conduct a study to determine if this hypothesis was supported empirically.

The results were obtained:

|Rat |Alone |Together |Difference |Sign of Difference |

| |(A) |(T) |= T - A | |

|1 |463 |523 | | |

|2 |462 |494 | | |

|3 |462 |461 | | |

|4 |456 |535 |79 | |

|5 |450 |476 |26 | |

|6 |426 |454 |28 | |

|7 |418 |448 |30 | |

|8 |415 |408 |-7 | |

|9 |409 |470 |61 | |

|10 |402 |437 |35 | |

What did we conclude regarding the research hypothesis?

Example 3: Swain vs. Alabama

In 1965, an appeal to the Supreme Court was made by a black man (the petitioner) sentenced to death after being convicted in the Circuit Court of Talladega County, Alabama, of the rape of a 17 year-old white girl by an all-white jury. At the time of the crime, the defendant was 19. The petitioner alleged (among other things) that the entire process of selecting eligible jurors, from the jury pool of eligible jurors to the jury panel (from which the trial jury is chosen), was racially discriminatory.

At the time, in the early 1960’s, eligible jurors in Alabama were males over 21. According to Census figures available then, there were 16,406 individuals that fit the profile in Talladega Country. Twenty-six percent of these individuals were African-Americans. When 100 individuals were chosen to serve on the jury panel, only 8 of them were African-Americans*. What is the likelihood of getting 8 blacks in a sample of 100 drawn from the population of males eligible for jury duty in Talladega County?

Example 4: Skin Cream

Pond’s Age-Defying Complex, a cream with alpha-hydroxy acid, advertises that it can reduce wrinkles and improve the skin. In a study published in Archives of Dermatology (June 1996), 33 women over age 40 used a cream with alpha-hydroxy acid for twenty-two weeks. At the end of the study, period 23 of women exhibited skin improvement (as judged by a dermatologist). Is this evidence that the cream will improve the skin of more than 60% of women over age 40?

Example 5: Hypertension in WSU students during finals week

Hypertension is defined as a blood pressure over 140 mmHg systolic and/or over 90 mmHg diastolic. Hypertension, if not corrected, can cause long-term health problems. In the college-age population (18-24 years of age), about 9.2% have hypertension. Suppose that a blood donor program is occurring in a college dormitory during finals week. Before each student gives blood, the nurse takes a blood pressure reading. Of 196 donors, it was found that 29 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than 9.2%?

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“k factorial” is the product of the numbers 1 through k and is denoted by k!

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