Consider a binomial distribution of 200 trials with expected...

Consider a binomial distribution of 200 trials with expected value 80 and standard deviation of about $6.9 .$ Use the criterion that it is unusual to have data values more than $2.5$ standard deviations above the mean or $2.5$ standard deviations below the mean to answer the following questions. (a) Would it be unusual to have more than 120 successes out of 200 trials? Explain. (b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain. (c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

Consider a binomial distribution of 200 trials with expected value 80 and standard deviation of about $6.9 .$ Use the criterion that it is unusual to have data values more than $2.5$ standard deviations above the mean or $2.5$ standard deviations below the mean to answer the following questions.
(a) Would it be unusual to have more than 120 successes out of 200 trials? Explain.
(b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain.
(c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

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Key Concepts

Tree Diagram

A tree diagram is a graphical tool used to represent all the possible outcomes of a sequence of events. Each branch represents a choice or outcome at every stage, making it easier to visualize and systematically list the sample space of a problem.

Sample Space

The sample space is the set of all possible outcomes of an experiment. In probability and combinatorics, accurately defining the sample space is key for determining the likelihood of events.

Counting Principle

The counting principle states that if there are several sequential events, the total number of outcomes is the product of the number of outcomes at each stage. This principle is used to calculate the size of the sample space.

Binomial Coefficient

The binomial coefficient quantifies the number of ways to choose a subset of outcomes (such as successes or specific outcomes) from a larger set when order does not matter. It is essential for calculating probabilities in binomial experiments.

Equally Likely Outcomes and Probability Calculation

When all outcomes in the sample space are equally likely, the probability of an event is determined by dividing the number of favorable outcomes by the total number of outcomes. This concept is fundamental in probability when analyzing discrete events.

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