Why Is It Not Possible In Example 8.1 On Page 256

8.3 Why is it not possible in Example 8.1 on page 256 to have 100% confidence? Explain.

In statistical inference, a confidence interval provides a range of values within which the true parameter (such as a population mean) is estimated to lie with a certain level of confidence. When we specify a 100% confidence level, we are asserting that the interval captures the true parameter with absolute certainty. However, in practice, constructing a 100% confidence interval is inherently problematic, which is why it is generally considered impossible or illogical to do so.

The fundamental reason is that confidence intervals are based on probability and the sampling distribution of the estimator. At less than 100% confidence levels (such as 95% or 99%), these intervals are associated with a specific probability that they contain the true population parameter when the process is repeated numerous times. These levels are mathematically defined based on the properties of the sampling distribution, such as the standard error and the critical value from the relevant statistical distribution (e.g., z-distribution or t-distribution).

When the confidence level approaches 100%, the critical value from the distribution tends toward infinity. For example, in the case of a normal distribution, the z-value for a 100% confidence interval would need to be infinite because the tail probability beyond any finite z-value must be zero. As a consequence, the interval would extend infinitely in both directions, encompassing all possible values. This renders the interval meaningless because it no longer provides any useful information about the parameter.

Furthermore, from an interpretative standpoint, a 100% confidence interval would imply absolute certainty. Since all statistical inference is based on sample data and the inherent randomness of sampling, it is impossible to guarantee with absolute certainty that the interval contains the true parameter. Variability in samples and potential biases mean that such a guarantee cannot be logically or practically achieved.

In the specific context of Example 8.1 on page 256, attempting to construct a 100% confidence interval would result in an unbounded, infinitely wide interval that includes all conceivable values, thus failing to provide any meaningful estimation. This highlights the fundamental limitation within the framework of statistical inference, emphasizing that confidence levels closer to 100% (but less than 100%) are always necessary to produce informative and practical confidence intervals.

References

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