Binomial Distribution

The binomial distribution is used to model the probability of a discrete random variable X. There are several conditions for the binomial to apply. There are a fixed number of trials, which must be independent from each other and each trial must follow a Bernoulli distribution (success/failure), with the same probability of success. The notation for a Binomial is given aswhere n is the number of trials, and p is the probability of success. The random variable X is number of successes for each trial. The individual values of X (individual elements in the sample space) are denoted with a k.

Calculating Probabilities

The probability mass function gives the probability when the random variable X is equal to a certain number (k). The k values are mutually exclusive, so to find probabilities of more than one k, we can add the values together. All of the probabilities of k must sum up to 1. The cumulative distribution function sums together all of the probabilities less than or equal to k. .

Question 1:

Birth Control Shot

The birth control shot is very effective method of birth control. In typical use, approximately 3% of women who use the shot become pregnant. We choose 20 women who have the shot and count the number who become pregnant in the following year. Use Minitab to calculate the PMF and CDF of the distribution and answer the questions.

1. Explain why this is a binomial distribution (how does it meet all of the criteria?)
2. What is the probability exactly 1 woman becomes pregnant?
3. What is the probability that at most one of the women become pregnant with typical use?
4. What is the probability that at least one of the women becomes pregnant with typical use?
5. Where is the 95th percentile in terms of number of pregnancies (hint: It's between two numbers)

Gaussian Distribution

The Gaussian distribution, often referred to as the normal distribution, is a density curve for specific sets of continuous data. These data sets are symmetrical and unimodal, meaning the mean and median are both located in the middle of the distribution. Most values in the population are near the mean. As the population values get further away from the mean, the number of subjects decreases. Most subjects in the population fall within three standard deviations from the mean. This type of distribution is very common with naturally occurring events. Human measurements: heights, weights, limb lengths etc.... all follow a normal distribution.

Finding the probability associated with a range of values for the Gaussian distribution is equivalent to finding the associated area underneath the density curve. A computer program or distribution calculator can find the area given the mean and standard deviation (parameters) of a normal distribution. A z-transform (converting all of the values to z scores), will convert the distribution to a standard normal , where the mean is equal to 0 and standard deviation equals 1. The standard normal distribution has the associated areas to the left of the curve. The z-transformation must be done in order to use the standard normal table, and then converted back to interpret the problem.

Question 2:

Soil pH

The pH of soil samples taken from a certain geographic region is normally distributed with a mean of 6 and a standard deviation of 0.15.

Draw the normal curve for this problem and use the 68-95-99.7 rule to label the axis.

Add the standard normal transformation to this curve. (hint: take the z-score for all values on your first curve).

Answer the following questions and provide a picture of the indicated probability.

1. Find the probability that the pH is greater than 6.10.
2. Find the probability that the pH is between 5.90 and 6.15
3. Find the 90th percentile of pH values.
4. Find the 32^ndpercentile of pH values.