Question

Problem 3: Celiac Disease

Celiac disease is an autoimmune disorder where the ingestion of gluten leads to damage in the small intestine. Left untreated, celiac disease can lead to the development of other autoimmune disorders like Type I diabetes, multiple sclerosis, anemia, and osteoporosis. Generally, the later in life that celiac disease is diagnosed, the higher the chances of developing another autoimmune condition. In fact, it is known that 34% of individuals with celiac disease that is first diagnosed when they are 21 years of age or older will develop another autoimmune condition. Suppose we are interested in the number of individuals that develop another autoimmune disorder in a random sample of 9 people with celiac disease first diagnosed after they turn 21. Assume these people are independent of each other.

1. Check if this situation fits the binomial setting. Write four complete sentences addressing each requirement in one sentence each.
2. Assuming this situation is a binomial experiment, build the probability distribution in table form in StatCrunch. There are two ways to do this. You may use Data à Compute à Expression and choose the function dbinom. This method relies on you entering the values of the random variable in the first column of your data table. The other way to do this is to use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 9. You may present this table horizontally or vertically and leave the probabilities unrounded.

3. Calculate the probability that exactly three people in the sample develop another autoimmune disorder using the StatCrunch binomial calculator. Copy this image from StatCrunch into your document. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

4. Calculate the probability that no more than 6 people in the sample develop another autoimmune disorder using the StatCrunch binomial calculator. Again, provide a StatCrunch binomial calculator graph to display your answer. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

5. Calculate the probability that between 2 and 5 people in the sample (inclusive) develop another autoimmune disorder. Show your work using the probability distribution you built in part (b) to answer this question. Then, verify it with a StatCrunch binomial calculator graph and include this image in your document as well. Finally, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

6. Calculate the mean and standard deviation of this probability distribution. Show your work using the binomial mean and standard deviation formulas and provide your answers in your document. (No need to use StatCrunch for this part).

Answer 1

a) To be a binomial probability distribution, a procedure must satisfy the preceding four requirements:

1. The procedure has a fixed number of trials. Here, we are interested in the number of individuals that develop another autoimmune disorder in a random sample of 9 people (fixed trials) with celiac disease first diagnosed after they turn 21.
2. The trials must be independent. Here, 9 people in the sample are independent because the disease of one does not affect the chance of disease of other people.
3. Each trial must have all outcomes classified into two categories. Here, responses were recorded into two categories, individuals with celiac disease that is first diagnosed when they are 21 years of age or older will develop another autoimmune condition and will not develop another autoimmune condition.
4. The probability of success remains the same in all trials. For all, it is known that 34% of individuals with celiac disease that is first diagnosed when they are 21 years of age or older will develop another autoimmune condition.

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b) In statcrunch, first enter values from 0 through 9 underVar1 column. Change Var1 name to x.

Click on Data->Compute->Expression.

Type dbinom(x,9,0.34) and click on compute.

The output is shown below:

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c) Click on Stat->Calculators->Binomial.

Fill values and press compute.

The output is shown below:

In the random sample of 9 people with celiac disease first diagnosed after they turn 21, the probability that exactly three people develop another autoimmune disorder is 0.2729.

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d) Click on Stat->Calculators->Binomial.

Fill values and press compute.

The output is shown below:

In the random sample of 9 people with celiac disease first diagnosed after they turn 21, the probability that atmost 6 people develop another autoimmune disorder is 0.9906.

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e) The mean and standard deviation of this probability distribution are: