Question

In developing an interval estimate, if the population standard deviation is unknown:

A) Use a t-distribution.
B) Use a z-distribution.
C) The estimate is not reliable.
D) Use the sample mean directly.

Answer 1

In developing an interval estimate, if the population standard deviation is unknown:A) Use a t-distribution.

Step-by-step explanation:

When developing an interval estimate and the population standard deviation is unknown, the appropriate distribution to use is the t-distribution. This is because the t-distribution accounts for the added variability introduced by estimating the population standard deviation with the sample standard deviation, s. The t-distribution is especially important when dealing with small sample sizes, where the normal (z) distribution might not provide accurate results.

The application of the t-distribution has become common practice when s is used as an estimate for σ (the population standard deviation). With access to modern computing tools such as graphing calculators and statistical software, the use of the t-distribution is now standard even when sample sizes are large. In this context, the correct answer to the question is option A) Use a t-distribution.

Answer 2

In developing an interval estimate when the population standard deviation is unknown, the appropriate choice is A) Use a t-distribution.

Step-by-step explanation:

When the population standard deviation is unknown, and the sample size is relatively small, it is advisable to use a t-distribution for developing an interval estimate. This is because the t-distribution accounts for the increased uncertainty introduced by using the sample standard deviation in place of the unknown population standard deviation. The larger the sample size, the closer the t-distribution approaches the standard normal distribution (z-distribution).

Option A correctly aligns with the statistical practice of utilizing the t-distribution in these circumstances. The t-distribution allows for a more accurate estimation of the population parameter by considering the variability introduced by using the sample standard deviation. This is crucial for constructing reliable interval estimates when dealing with small sample sizes and unknown population standard deviations.

In summary, the choice of using a t-distribution reflects a statistical best practice when developing interval estimates in situations where the population standard deviation is unknown. The t-distribution provides a more realistic representation of the sampling variability under these conditions, ensuring the reliability of the interval estimate.