stats - binomial probabilities

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Why Study Binomial Distributions? Remember that, although statistical inference takes many forms, it often follows these phases: 1. Identify what you want to study. 2. Ensure that your study and your sample are valid and useful for drawing conclusions

about what you're studying. This includes understanding how the data were collected (so you know their limitations).

3. Recognize the type of probability distribution that models your situation. The

probability distribution tells you how likely or unlikely your findings are. You can also use it to find the range of values your population parameter is likely to have.

4. If you're comparing at least two measures or studying a relationship, conduct

significance tests to compare your results against what you'd expect from chance alone. If the study's results are highly unlikely, your results are statistically significant.

This Activity focuses on phase 3, the probability distribution, with particular emphasis on the binomial distribution. You'll need to know about a few different types of probability distributions to do inferential statistics. Review Distribution is one of the most important concepts in statistics. Every variable has a distribution, whether the variable is categorical or numerical, continuous or discrete. A variable's distribution consists of the values it takes and how often it takes each of these values. Distributions are often shown in graphs so you can see the relative frequency with which values occur. A binomial distribution (also referred to sometimes as the binomial setting) is a discrete probability distribution of the counts of successes and failures, such as the number of times you get heads when you flip a coin 100 times. Most textbooks list the following four characteristics for the binomial distribution (or setting): In the Binomial Setting, 1. There's a set n of identical trials. 2. The outcome of each trial is either success or failure. 3. The probability of success of each trial is p. (The probability of failure of each trial is

q = 1 - p.) 4. Each trial is independent.

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AP Statistics Study Sheet: Binomial Settings and Binomial Probabilities

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Some textbooks list the following as a fifth characteristic, and others use it as an introduction to the previous four:

5. In the binomial setting, we're interested in x, the number of successes observed during the n trials, for x = 0, 1, 2, 3 , , n.

The following are examples of binomial settings: What's the probability of getting exactly three heads on five flips of a coin? A coin is flipped 100 times. What's the probability of getting 45 heads? If the probability of having Rh-negative blood is .4 in a population of 500, what's the

probability 210 have type Rh-negative blood?

These are all probability settings in which: 1. We know the number of repetitions. 2. The outcome of each trial is either success or failure. 3. We know the probability of success or failure of any trial. 4. The probability doesn't change from trial to trial (the trials are independent.) Other Definitions to Remember Binomial Event: a set of trials within a binomial setting Simple Event or Outcome: a single trial B(n, p): the binomial probability distribution with parameters n, the number of observations, and p, the probability of success on any one observation. If X is a binomial

random variable with B(n, p), B(n, p, x) = P(X = x) =

xn

(p)x(1 p)nx.

On your calculator, binompdf(n,p,X) calculates the probability of exactly x successes in n trials with a probability of p. With a range of outcomes for X, where X takes on the number of successes 0, 1, 2, 3, , n, use binomcdf(n,p,X) on your calculator, which calculates the probability of up to and including x successes in n trials with a probability of p for the binomial probability distribution. Remember, the binomial distribution is discrete: Each single value has a probability. This means that, unlike a continuous distribution such as the normal distribution, there's a difference between P(x < 4) and P(x 4). So for P(x < 4), you add P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3). But for P(x 4) you include the 4, so you add P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4). Be careful about your endpoints when using binomcdf. With binomcdf on a calculator, you get the lower-tail probability that includes the upper bound. So, if you enter binomcdf(20,.3,5), you're finding P(x 5) for B(20, .3). If you need a range with a specific lower and upper bound, remember to subtract like this: (probability below upper bound) (probability below lower bound).



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Be careful about the range you calculate. It may help to draw the range on a number line. For example, if you want to calculate P(4 < x < 8) for B(10, .3), you want the range represented by the underlined numbers: 0 1 2 3 4 5 6 7 8 9 10. So you'd calculate: binomcdf(10,.3,7) binomcdf(10,.3,4). Note that this will give you the range from 5 to 7. If you wanted to calculate P(4 x < 8), you include the 4 in your range: 0 1 2 3 4 5 6 7 8 9 10. So you'd calculate: binomcdf(10,.3,7) binomcdf(10,.3,3). Note that now you're subtracting the 3 (since it's included in binomcdf(10,.3,3)) but not the 4. If you want P(4 x 8), you'd include the 8: 0 1 2 3 4 5 6 7 8 9 10. You'd calculate: binomcdf(10,.3,8) binomcdf(10,.3,3). There are cases where the criteria of the binomial setting might not be satisfied, but which would still be considered binomial. An event is almost binomial if the population size of an experiment is at least 20 times larger than the sample size. (NOTE: 20 is a rule of thumb. Some references use 10 instead of 20.)



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