For very large samples, the z-distribution can be used even if the population standard deviation is unknown.

For very large samples, the z-distribution can be used even if the population standard deviation is unknown.

• A. The z-distribution should always be used regardless of sigma.
• B. The t-distribution should never be used with large samples.
• C. For very large samples, the z-distribution can be used even if sigma unknown.
• D. The choice has no impact on CI width.

Answer: (C)

For very large samples, using the z-distribution even when the population standard deviation is unknown works because the distribution of the sample mean becomes highly stable and well-approximated by a normal curve as n grows. When sigma isn’t known, you typically estimate the standard error with s/√n and would use the t-statistic. However, the t distribution with df = n−1 converges to the standard normal distribution as the degrees of freedom increase. That means the critical values from t and z become nearly identical and the resulting confidence intervals (or hypothesis tests) are virtually the same for large n. So the z approach is a good approximation in this context. In contrast, for small samples the t-distribution is preferred because it has heavier tails and more accurately accounts for the additional uncertainty in estimating sigma. The choice does affect the width of the CI in finite samples—the z interval is typically narrower than the t interval when sigma is unknown and n isn’t large enough.

For very large samples, using the z-distribution even when the population standard deviation is unknown works because the distribution of the sample mean becomes highly stable and well-approximated by a normal curve as n grows. When sigma isn’t known, you typically estimate the standard error with s/√n and would use the t-statistic. However, the t distribution with df = n−1 converges to the standard normal distribution as the degrees of freedom increase. That means the critical values from t and z become nearly identical and the resulting confidence intervals (or hypothesis tests) are virtually the same for large n. So the z approach is a good approximation in this context.

In contrast, for small samples the t-distribution is preferred because it has heavier tails and more accurately accounts for the additional uncertainty in estimating sigma. The choice does affect the width of the CI in finite samples—the z interval is typically narrower than the t interval when sigma is unknown and n isn’t large enough.