Question
In a community, 65% of households have at least one college graduate. A random sample contains 100 households. Let X count the sampled households with at least one college graduate. Find the mean of X.
Answer
The expected value is 65 households.
The count is modeled as , where:
| Parameter | Value | Meaning |
|---|---|---|
| 100 | Number of sampled households | |
| 0.65 | Probability a household has at least one college graduate | |
| 0 through 100 | Number of sampled households meeting that condition |
For a binomial count, the expected-value formula is
Interpret the result
Across repeated samples under this model, the average count would approach 65 qualifying households per sample of 100. An individual sample can contain fewer or more than 65; the expected value is not a guaranteed observed count.
Use 0.65, not 65, for the success probability. The answer is a count of households, whereas is the expected sample proportion.
Why the parameters matter
A binomial distribution requires a fixed trial count and a common success probability, with independent trials for the binomial model. Here, the textbook supplies the trial count and success rate in words, so no missing plot needs to be reconstructed. Random sampling from a finite population without replacement is not exactly independent; treating the sample as binomial assumes that dependence is negligible or that independent sampling is intended.
Evidence boundary
This answers OpenStax Practice Test 2, question 22 with its shared household stem: n=100 and p=0.65. It is a complete text-based representative exercise, not a reconstruction of an unspecified graph. The binomial assumption is stated explicitly; 65 is an expectation, not an observed result.
Sources
These references support the concepts and methods used in the explanation above.