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Expected value is a fundamental concept in probability that helps predict the average outcome of a random event over time. It effectively tells us what we "expect" to happen on average when we conduct an experiment multiple times. For a binomial distribution, which counts successes in a sequence of yes/no experiments, the expected value can be calculated by multiplying the number of trials by the probability of success in each trial.

In the exercise provided, the number of trials is the sample size of 120, and the probability that an American adult had chickenpox before adulthood is 0.9 (or 90%). Thus, the expected value, denoted as \(E(X)\), is computed as follows:

• \(E(X) = n \cdot p = 120 \times 0.90 = 108\)

This indicates that in a sample of 120 adults, you would expect 108 of them, on average, to have had chickenpox.

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation means the values are close to the mean, whereas a high standard deviation indicates a wider spread. In binomial distributions, it quantifies how much the number of successes in a set of trials could typically deviate from the expected value.

To compute the standard deviation in a binomial distribution, the formula used is:

• \(\sigma = \sqrt{n \cdot p \cdot (1-p)}\)

For the exercise:

• \(\sigma = \sqrt{120 \cdot 0.90 \cdot 0.10} = \sqrt{10.8} \approx 3.29\)

This tells us that while we expect 108 people to have had chickenpox, it's typical for the actual number to vary by about 3.29 people.

The z-score helps determine how far away a given data point is from the mean of the distribution, expressed in standard deviations. It essentially tells us how typical or atypical our observation is compared to the average. A z-score near 0 indicates the data point is close to the mean, whereas higher absolute values imply that it is further away.

In the given exercise, the z-score was used to gauge whether observing 105 people who had chickenpox is surprising compared to the expected 108. The z-score is calculated as:

• \(z = \frac{X - E(X)}{\sigma} = \frac{105 - 108}{3.29} \approx -0.912\)

A z-score of approximately -0.912 is within one standard deviation from the mean, indicating that the observation is not surprising or unusual.

A binomial distribution is a type of probability distribution that summarizes the likelihood of a value occurring in a fixed number of independent experiments, each with the same probability of success. It's particularly useful when dealing with yes/no type problems. It is characterized by two key parameters: number of trials \(n\) and probability of success \(p\).

In our chickenpox example, the distribution helps us understand the likelihood of various outcomes, such as having exactly 105 people in the sample who have had chickenpox. To determine how likely a certain number of successes is (like 105), we can calculate the cumulative probability \( P(X \leq 105) \). This reflects the chance of observing 105 or fewer successes:

• \(P(X \leq 105) \approx 0.184\)

This indicates that there is an 18.4% likelihood of having 105 or fewer chickenpox cases, showing that while not certain, it is a reasonable outcome within this statistical model.