Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.3.9c

Chapter 5, Problem 5.3.9c

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births?

Verified step by step guidance

Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The formula for the Poisson probability is P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events, λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Calculate the average number of births per day (λ). Since there were 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.

Step 3: Use the Poisson formula to calculate the probability of no births in a single day (k = 0). Substitute k = 0 and the calculated value of λ into the formula: P(X = 0) = (λ^0 * e^(-λ)) / 0!. Note that 0! = 1 and λ^0 = 1, so the formula simplifies to P(X = 0) = e^(-λ).

Step 4: Determine whether 0 births in a single day is a significantly low number. A result is typically considered significantly low if its probability is less than or equal to 0.05. Compare the calculated probability P(X = 0) to this threshold to make a conclusion.

Step 5: Interpret the result. If P(X = 0) is less than or equal to 0.05, then 0 births in a single day would be considered significantly low. Otherwise, it would not be considered significantly low. Provide reasoning based on the calculated probability and the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which provides the expected number of births per day.

Significance of Events

Determining whether an event is significantly low involves comparing the observed outcome to the expected outcome under the Poisson distribution. In this case, if the probability of observing 0 births in a day is very low (typically below a threshold like 0.05), it may be considered significantly low, indicating that such an occurrence is unusual given the average rate of births.

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Textbook Question

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c. Find the probability of 40 or more first lines for Democrats.

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Textbook Question

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

c. Find the probability of 152 or more yellow peas.

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Textbook Question

d. Which probability is relevant for determining whether 40 first lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 first lines for Democrats significantly high?

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