## Statistics - Binomial Distribution

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This activity consists of a single scenario-based problem allowing students to apply the binomial probability distribution to decide whether or not an outcome is likely binomial distribution statistics problems. Summary This activity consists of a single scenario-based problem allowing students to apply the binomial probability distribution to decide whether or not binomial distribution statistics problems outcome is likely random.

Students will be able to apply the binomial probability distribution in a realistic setting. This activity is appropriate for a introductory level business statistics course. It is appropriate for use in any size class. It can be binomial distribution statistics problems either to introduce the concepts of the binomial distribution, or to assess students' understanding after the binomial distribution has been discussed.

If the problem is done in class it will take about minutes to complete. You are watching an episode of Law and Order. The plot for this episode includes a scientist that was trying to communicate with patients in a persistent vegetative state.

She put two signs in front of the patients. The sign on the right side of the patient read 'Yes. In the show there is a disagreement over whether or not the scientist was actually communicating with these patients, or if she was simply recording random eye movements in these patients. The scientist claimed the percent of correct responses she got showed she was able to communicate with these patients. What percent of correct responses do you believe she should have gotten in order to claim she was able to communicate with the patients?

This activity can be done in class individually or in small groups. It can also be assigned as a homework problem or as an exam question.

Students need to have some familiarity with the binomial probability distribution to successfully complete the problem. If your students are new to context-rich problems, you may want to include prompts to help students, binomial distribution statistics problems as 'What probability distribution binomial distribution statistics problems best suited to a situation like this one?

You should expect to see students attempting to apply a wide variety of probability rules. Class discussion can focus on what types of questions are answered with the probability rules they have applied and how these questions differ from the one posed. The purpose of the assessment will determine whether or not you need a rubric.

If the problem will be graded, it may be helpful to give the students a rubric such as: All statistical reasoning in the answer is correct. All relevant binomial distribution statistics problems are included. May have minor mistakes, such as a minor error in a calculation. Statistical reasoning in the answer is correct, but the answer contains minor mistakes such as calculation errors. Contains significant errors in the statistical reasoning, such as missing steps in the problem solving process, or several significant errors in calculations.

Very little of the statistical reasoning is correct and relevant to the problem. None binomial distribution statistics problems the economic content is relevant to the question. Teaching and Learning Economics Starting Point: Why Teach with Context-Rich Problems? June 02, Printing Shortcut:

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A Binary categorical variable is a variable that has two possible outcomes. Let's use the example from the previous page investigating the number of prior convictions for prisoners at a state prison at which there were prisoners.

A special discrete random variable is the binomial. We have a binomial experiment if ALL of the following four conditions are satisfied:. In general, we see the mean of a binomial is the number of trials times the probability of success.

The standard deviation is the square root of the mean times the probability of failure. Suppose that in your town 3 such crimes are committed and they are each deemed independent of each other. First, we must determine if this situation satisfies ALL four conditions of a binomial experiment stated above:.

To do this we find the probability that one of the crimes would be solved. With three such events crimes there are three sequences in which only one is solved.

We add these three probabilities up and get 0. Looking at this from a formula standpoint, we have three possibile sequences, each involving one solved and two unsolved events. Putting this together gives us the following:. The factorial of a number means to take that number and multiply it by every number that comes before it - down to one excluding 0. What is the probability that at least one of the crimes will be solved? This would be to solve: OR, we could simplify our work by using the complement rule.

We have carried out this solution below. In such a situation where three crimes happen, what is the expected number of crimes that remain unsolved and the standard deviation? Here we are applying the formulas from above. Below is another example in which we illustrate how to use the formula to compute binomial probabilities again. Now we cross-fertilize five pairs of red and white flowers and produce five offspring.

Find the probability that there will be no red flowered plants in the five offspring. Now, find the probability that there will be four or more red flowered plants. Try to figure out your answer first, then click the graphic to compare answers. The mean of a distribution is also called the expected value of the distribution. Of the five cross-fertilized offspring, how many red flowered plants do you expect?

Y can only take values 0, 1, 2, What is the standard deviation of Y , the number of red flowered plants in the five cross-fertilized offspring? The treatment was tried on 40 randomly selected cases and 11 were successful. Do you doubt the company's claim? So we know the binomial is approximately normal. So here we have n equals 40 and the probability of success pi of 0. So these two are true. So that means we can use normal approximation methods or the empirical rule.

So, recall from the emprical rule that we would expect 68 percent of observations within one standard deviation, 95 percent within two standard deviations, and So our mean here for the binomial is equal to n time pi which would be equal to 40 times 0.

Then we know our standard deviations for our binomial would be equal to n time pi times one minus pi, then take the square root of that, so now we have 40 times 0. This gives us 24 plus or minus 3 times 3. This results in a range of So, we would expect almost all of the counts out of 40 to be some where between And the question is, "Is getting 11 unlikely? So, this is an application of the empirical rule to the binomial when we have the approximate normal distribution of a binomial under these two conditions.

If the probability is large, do not doubt the claim. If the probability is small, doubt the claim. Using Minitab, we get the following output:. The probability is very small. We, thus doubt the claim. It is incorrect to just compute the probability at 11 since that is usually very small if sample size is large. Andrew Wiesner again, working through this alternative approach:. How are probability values and cumulative probability values related?

This is an important relationship to understand. Alternatuively, and this can be used in any situation, so it doesn't necessairly have to be a normal approximation, so the prior example was relevant only when we met the normal approximation, but this will be use any time. In this approach, here what we are saying is that we have 40 as our trial or sample size and we have probailit of success of 0.

Our notation here, we have used this as our notation Y , but we will see where Minitab uses this X. This is just the Mintab notation. There is no difference between the two, it is just a matter of preference. When we do this in Minitab, we see we get 0.

When we add all of this up, we end up with this probability of 0. So, obviously this is extremely unlikely. So, the chance of getting 11 or fewer successes under the situation where n is equal to 40 and the probabilty of success is equal to 0. So therefore, it would be extremely unlikely in this situation that you would have 11 or fewer successes.

This is an alternative to the empirical rule, which again you can use in any situation, whereas the empirical rule would apply in situations where the binomial is approximately normal. Eberly College of Science. Printer-friendly version Unit Summary. First, we must determine if this situation satisfies ALL four conditions of a binomial experiment stated above: Does it satisfy fixed number of trials? Does it have only 2 outcomes? YES Stated in the description.

Putting this together gives us the following: If we fill in the formula above using the data from our example it would be: Find the probability that there will be four or more red flowered plants.

What is the standard deviation of Y? Both are at least 5. Welcome to STAT ! A Note on Notation!