If you constructed one hundred 95% confidence intervals based on one hundred different simple random samples of size n, how many of the intervals would you expect to include the unknown parameter? Assume all model requirements are satisfied.

Verified step by step guidance

1. Understand that a 95% confidence interval means that if we were to take many samples and construct confidence intervals in the same way, approximately 95% of those intervals would contain the true population parameter.
2. Recognize that the problem states you have 100 different simple random samples, each producing a 95% confidence interval for the unknown parameter.
3. Calculate the expected number of intervals that include the true parameter by multiplying the total number of intervals by the confidence level: \(100 \times 0.95\).
4. Interpret this result as the expected count of intervals (out of 100) that will contain the unknown parameter, assuming all model assumptions are met and the sampling process is correct.
5. Note that this is an expectation, so the actual number of intervals containing the parameter may vary in practice, but on average, it should be close to this value.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter. It provides an estimated range with a specified confidence level, such as 95%, indicating the degree of certainty in the estimate.

Confidence Level

The confidence level represents the proportion of confidence intervals, constructed from repeated samples, that are expected to contain the true population parameter. For a 95% confidence level, about 95 out of 100 such intervals will include the parameter.

Simple Random Sampling

Simple random sampling is a method where each member of the population has an equal chance of being selected. This ensures that samples are representative and that the assumptions underlying confidence interval calculations are valid.