Final answer:
When the standard deviation is not provided in sample size calculations, statisticians use the sample standard deviation as an estimate, often using statistical software or calculators like the TI-83. In small samples, the Student's t-distribution is utilized assuming a normally distributed population. The use of technology is recommended due to the complexity of these calculations.
Step-by-step explanation:
When calculating sample size and the standard deviation is not provided, other statistical methods must be employed since the standard deviation is a crucial component in calculation. If the population standard deviation is unknown, which is common in practice, statisticians use the sample standard deviation (s) as an estimate for the population standard deviation (σ) and adjust their calculations accordingly. This is often done using technology such as statistical software or calculators like the TI-83, which can be tedious without technological aid.
In scenarios of small sample sizes, statisticians use the Student's t-distribution, assuming the underlying population is normally distributed. Remember, while performing calculations such as these, intermediate results should not be rounded off to maintain accuracy. The standard error of the difference between two sample means, X₁ - X₂, can also be calculated using the sample standard deviations from the independent samples.
For example, if dealing with a single sample from a population, you could calculate the value that is one standard deviation above the mean by using the formula x = x + (#ofSTDEVs)(s) where x is the sample mean and s is the sample standard deviation. The correct calculator or computer software settings must be selected, such as ox or sx from the summary statistics, to ensure accurate computations. These tools help to compute the various statistics like the mean and standard deviation from the sample data efficiently.