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Probability calculations are the backbone of many statistical analyses and are essential to understand the likelihood of different outcomes. In this exercise, the scenario represents a binomial distribution. Each American asked whether they believe the country is in a recession can say either "yes" or "no." This follows a Bernoulli trial, where each trial has two possible outcomes with a fixed probability.

• Our probability of "success" (believing in a recession) is denoted by \( p = 0.5 \), meaning there is a 50% chance a person will believe it's a recession.
• The number of trials, or sample size, is \( n = 20 \).
• The probability of exactly \( k \) successes (believing it's a recession) in \( n \) trials is calculated using the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] To compute precisely 12 people (part a), plug \( n = 20 \), \( k = 12 \), and \( p = 0.5 \) into the formula. For no more than 5 people (part b), sum the individual probabilities for \( X \leq 5 \). This involves calculating repeatedly for \( k = 0, 1, 2, 3, 4, \) and \( 5 \), before summing them up.

The expected value in a binomial distribution gives the average outcome or the "center" of the distribution. For example, if you were to repeat the process of asking 20 people these same questions many times, the expected value tells you what the average number of people would say "yes."
The formula to determine the expected number of people who believe in the recession is:
\[ E(X) = np \] Using \( n = 20 \) and \( p = 0.5 \), the calculation becomes:
\[ E(X) = 20 \times 0.5 = 10 \] This calculation reveals that on average, 10 people out of every 20 sampled are expected to believe that the country is in a recession. This value, being the midpoint of the distribution, helps us understand the overall tendency or expected outcome in this process. It's a theoretical measure signifying the mean of the probability distribution.

Variance and standard deviation help us understand how much the responses are expected to deviate from the expected value. While the expected value gives us the "average," variance and standard deviation tell us about the "spread" or variability.

• Variance is computed using the formula:

\[ Var(X) = np(1-p) \] For our problem, substituting \( n = 20 \) and \( p = 0.5 \), we find:
\[ Var(X) = 20 \times 0.5 \times (1-0.5) = 5 \]

• Standard Deviation is simply the square root of the variance:

\[ \sigma = \sqrt{Var(X)} = \sqrt{5} \approx 2.236 \] These measures tell us that while we expect an average of 10 people to believe in a recession, the number usually deviates by about 2.236 people. This understanding allows statisticians to estimate the confidence and reliability of the predictions made from the probability distribution.