Final answer:
Given a sample size of 38 from a population with a known standard deviation, it is appropriate to use the normal distribution according to the Central Limit Theorem since the sample size is larger than 30.
Step-by-step explanation:
When considering whether to use the normal distribution to find probabilities for the sample mean, the Central Limit Theorem (CLT) provides the guidance needed. For a sample of size 38 drawn from a population with a known mean (μ = 48) and standard deviation (σ = 10), the CLT suggests that if either the population itself is normally distributed or the sample size is sufficiently large (usually n ≥ 30 is considered large enough), then the distribution of the sample means will approximate a normal distribution.
In this scenario, since the sample size of 38 is greater than 30, and no information suggests the population is not normally distributed, it is appropriate to use the normal distribution to estimate the probabilities for the sample mean.
However, if we were testing a sample mean without a known population standard deviation, a Student's t-distribution would be more appropriate. This does not apply here as the population standard deviation is given. To perform these calculations on a TI-84 Plus calculator, one would typically use functions that utilize the sample mean, the population standard deviation, and the sample size to calculate probabilities or confidence intervals.