Long Life? In a poll conducted by Genworth Financial, a random sample of adults was asked, “What age would you like to live to?” The results of the survey are given in the table.
a. Construct a probability model of the data.

Verified step by step guidance

1. Step 1: Identify the total number of respondents by summing all the frequencies given in the table. This means adding 126 + 262 + 263 + 388 to find the total sample size.
2. Step 2: Understand that a probability model assigns probabilities to each age group based on the relative frequency of respondents who chose that group. The probability for each age group is the number of respondents in that group divided by the total number of respondents.
3. Step 3: Calculate the probability for each age group using the formula: \(P(\text{age group}) = \frac{\text{Number in age group}}{\text{Total number of respondents}}\).
4. Step 4: Write the probability model as a list or table showing each age group paired with its corresponding probability calculated in Step 3.
5. Step 5: Verify that the sum of all probabilities equals 1, which confirms that the probability model is valid.

Key Concepts

Probability Model

A probability model assigns probabilities to all possible outcomes of a random experiment. In this context, it involves converting the counts of people preferring each age range into probabilities by dividing each count by the total number of respondents. This model helps describe the likelihood of each age preference.

Relative Frequency

Relative frequency is the ratio of the number of times an event occurs to the total number of trials or observations. Here, it is used to estimate the probability of each age group by dividing the number of respondents in that group by the total sample size, providing an empirical probability distribution.

Random Sampling

Random sampling means selecting individuals from a population in such a way that each member has an equal chance of being chosen. This ensures the sample represents the population fairly, allowing the probability model based on the sample data to be generalized to the larger population.