Questions: Determine if the following probability experiment represents a binomial experiment. If not, explain why. If the probability experiment is a binomial experiment, state the number of trials, n, and probability of success, p. An investor randomly purchases 13 stocks listed on a stock exchange. Historically, the probability that a stock listed on this exchange will increase in value over the course of a year is 43%. The number of stocks that increase in value is recorded.

Solution Steps

The probability experiment involves an investor randomly purchasing 13 stocks and recording the number of stocks that increase in value. This experiment meets the criteria for a binomial experiment because:

1. There are a fixed number of trials, \( n = 13 \).
2. Each trial (stock purchase) has two possible outcomes: success (the stock increases in value) or failure (the stock does not increase in value).
3. The probability of success, \( p = 0.43 \), remains constant for each trial.
4. The trials are independent of each other.

Using the parameters of the binomial experiment, we can calculate the following statistical measures:

• Mean \( \mu \): \[ \mu = n \cdot p = 13 \cdot 0.43 = 5.59 \]

• Variance \( \sigma^2 \): \[ \sigma^2 = n \cdot p \cdot q = 13 \cdot 0.43 \cdot (1 - 0.43) = 3.1863 \]

• Standard Deviation \( \sigma \): \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{13 \cdot 0.43 \cdot 0.57} = 1.785 \]

The results of the analysis are as follows:

• Mean (expected value): \( \mu = 5.59 \)
• Variance: \( \sigma^2 = 3.1863 \)
• Standard Deviation: \( \sigma = 1.785 \)

Final Answer