Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.
Fear of Heights The table lists results from a survey of 285 subjects who were asked, “Are you afraid of heights in tall buildings?” The results are from USA Today.

Verified step by step guidance

1. Step 1: Verify if the given table represents a probability distribution. To do this, check two key requirements: (a) All probabilities P(x) must be between 0 and 1, and (b) The sum of all probabilities P(x) must equal 1.
2. Step 2: Add the probabilities provided in the table: P(Yes) = 0.46 and P(No) = 0.54. Use the formula: ∑P(x) = P(Yes) + P(No).
3. Step 3: If the sum of probabilities equals 1 and all probabilities are between 0 and 1, confirm that the table represents a valid probability distribution. If not, identify which requirement is violated.
4. Step 4: To find the mean (expected value) of the distribution, use the formula: E(X) = ∑[x * P(x)], where x represents the response values (assign numerical values such as Yes = 1, No = 0) and P(x) represents the probabilities.
5. Step 5: To find the standard deviation, use the formula: σ = √∑[(x - E(X))² * P(x)], where E(X) is the mean calculated in Step 4, and x and P(x) are the response values and probabilities respectively.

Key Concepts

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. For a valid probability distribution, the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1. In the given survey results, the probabilities for 'Yes' and 'No' responses sum to 1, indicating a valid probability distribution.

Mean of a Probability Distribution

The mean, or expected value, of a probability distribution is a measure of the central tendency of the random variable. It is calculated by multiplying each possible value by its corresponding probability and summing these products. In this case, the mean would provide insight into the average response regarding fear of heights based on the survey data.

Standard Deviation of a Probability Distribution

The standard deviation measures the dispersion or spread of a probability distribution around its mean. It quantifies how much the values of the random variable deviate from the expected value. A low standard deviation indicates that the values are close to the mean, while a high standard deviation suggests a wider spread of responses, which can be calculated using the probabilities and their respective outcomes.