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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblema 4.Q.4b
Capitolo 4, Problema 4.Q.4b

The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)
b. Graph the binomial distribution using a histogram and describe its shape.

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Step 1: Understand the problem. This is a binomial distribution problem where the probability of success (surviving 5 years after a liver transplant) is 0.75, and the number of trials (patients) is 6. The goal is to graph the binomial distribution and describe its shape.
Step 2: Recall the formula for the binomial probability mass function (PMF): P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success, and (n choose k) = n! / [k! * (n-k)!]. Use this formula to calculate the probabilities for k = 0, 1, 2, ..., 6.
Step 3: Create a table of values for k (number of successes) and their corresponding probabilities P(X = k). For each value of k, substitute into the binomial PMF formula to compute the probability.
Step 4: Plot a histogram using the values of k on the x-axis (0 to 6) and the corresponding probabilities P(X = k) on the y-axis. Each bar in the histogram represents the probability of a specific number of successes.
Step 5: Analyze the shape of the histogram. Since the probability of success (0.75) is relatively high, the distribution is expected to be skewed slightly to the left, with the highest probabilities concentrated around higher values of k (e.g., 4, 5, 6).

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Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, the success is defined as a patient surviving five years post-liver transplant, with a probability of 0.75. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Mean & Standard Deviation of Binomial Distribution

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. Each bin's height reflects the frequency of data points within that interval. In the context of the binomial distribution, the histogram will display the number of patients who survive versus those who do not, illustrating the probabilities of different outcomes.
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Intro to Histograms

Shape of the Distribution

The shape of a binomial distribution can vary based on the probability of success and the number of trials. When the probability of success is greater than 0.5, the distribution is typically skewed to the right, indicating more successes. In this scenario, with a 75% survival rate and six patients, the histogram is expected to show a right-skewed shape, reflecting a higher likelihood of more patients surviving.
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