Final answer:
To calculate the probability of X being less than 94 for a sample of size n=16 from a normal distribution with mean 100 and standard deviation 8, we compute the standard error, convert the value to a z-score, and then find the corresponding probability from standard normal distribution, resulting in approximately 0.13%.
Step-by-step explanation:
To find the probability that X is less than 94, we first need to determine the standard error of the mean since we have a sample size (n=16) from a population with a mean (μ) of 100 and a standard deviation (σ) of 8. The standard error (SE) is calculated by the formula SE = σ/sqrt(n). In this case, SE = 8 / sqrt(16) = 8 / 4 = 2.
Now we can standardize the sample mean score of 94 to a z-score using the formula z = (X - μ)/SE, where X is the sample mean we want to compare. So z = (94 - 100) / 2 = -6 / 2 = -3.
Using the standard normal distribution, we look up the probability of z being less than -3. From standard normal distribution tables or using a calculator with normal distribution functions, we find the corresponding probability. For z = -3, the probability P(X < 94) is approximately 0.0013 or 0.13%.