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The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC College has a student advisory committee made up of ten staff members and six students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? The names of all committee members are put into a box, and two names are drawn without replacement . The first name drawn determines the chairperson and the second name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student on the first draw is $\frac{6}{16}$ . The probability of a student on the second draw is $\frac{5}{15}$ , when the first draw selects a student. The probability is $\frac{6}{15}$ , when the first draw selects a staff member. The probability of drawing a student's name changes for each of the trials and, therefore, violates the condition of independence.
A lacrosse team is selecting a captain. The names of all the seniors are put into a hat, and the first three that are drawn will be the captains. The names are not replaced once they are drawn (one person cannot be two captains). You want to see if the captains all play the same position. State whether this is binomial or not and state why.
This is not binomial because the names are not replaced, which means the probability changes for each time a name is drawn. This violates the condition of independence.
“Access to electricity (% of population),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc (accessed May 15, 2015).
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A statistical experiment can be classified as a binomial experiment if the following conditions are met:
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np , and the standard deviation is given by the formula σ = $\text{}\sqrt{npq}$ .
X ~ B ( n , p ) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p .
X = the number of successes in n independent trials
n = the number of independent trials
X takes on the values x = 0, 1, 2, 3, ..., n
p = the probability of a success for any trial
q = the probability of a failure for any trial
p + q = 1
q = 1 – p
The mean of X is μ = np . The standard deviation of X is σ = $\sqrt{npq}$ .
Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.
In words, define the random variable X .
X = the number that reply “yes”
X ~ _____(_____,_____)
What values does the random variable X take on?
0, 1, 2, 3, 4, 5, 6, 7, 8
Construct the probability distribution function (PDF).
x | P ( x ) |
---|---|
On average ( μ ), how many would you expect to answer yes?
5.7
What is the standard deviation ( σ )?
What is the probability that at most five of the freshmen reply “yes”?
0.4151
What is the probability that at least two of the freshmen reply “yes”?
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