Final answer:
The correct assumption necessary to perform hypothesis testing using a Student's t-distribution is that the distribution of the differences between measurements must be approximately normal, as stated in option A.
Step-by-step explanation:
When conducting hypothesis testing, particularly when using a Student's t-distribution, it is crucial to make certain assumptions about the population from which the sample data has been drawn. One essential assumption is that the distribution of the differences between the measurements must be approximately normal. This assumption allows the use of the t-distribution for the statistical analysis, ensuring that the results are valid under the t-test model.
The requirement of normality is based on the central limit theorem, which states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal regardless of the population's actual distribution. However, if the sample sizes are small, it is especially important that the actual distribution of the data is normal for the results to be reliable.
Hence, option A is the assumption necessary to perform the test in question. It presumes that the population distribution of the differences should be approximately normal. This aligns with the assumptions detailed earlier in the course for performing tests of a population mean using the normal distribution or Student's t-distribution.