Final answer:
To find probabilities within a normal distribution, calculate the Z-scores for the given values, use a standard normal distribution table or calculator functions like normalcdf and invNorm, and understand the central limit theorem's effect on sampling distributions.
Step-by-step explanation:
Finding Probability in Normal Distribution
To find the probability of a sample mean being between 85 and 92 from a normal distribution with a mean (μ) of 90 and a standard deviation (σ) of 15 for samples of size n = 25, we utilize the central limit theorem. The sampling distribution of the sample mean will also be normally distributed with mean μ and standard deviation σ/√n, where n is the sample size. Thus, the standard deviation of the sampling distribution, often called the standard error, is 15/√25 = 3. The Z-scores for 85 and 92 are (85-90)/3 and (92-90)/3 respectively. Finally, we look up these Z-scores in the standard normal distribution table or use a calculator with normal distribution functions to find the probabilities.
Regarding the area above 138 for a normal distribution of N(160,25), we convert the value 138 into a Z-score using the formula Z = (X - μ)/σ, and then use a standard normal distribution table to find the area to the right of this Z-score. Similarly, to calculate the area to the right for any given Z-score, you subtract the area to the left from 1. For cumulative distributions, such as Poisson or normal distributions with non-standard parameters, we typically use a calculator or software to find these probabilities, as in the example of P(X ≤ 160) with a normal approximation.
For determining percentile values or probabilities of a range in a normal distribution, we can utilize functions like invNorm to find the kth percentile for a normally distributed variable, or normalcdf for cumulative probabilities over a range. It's crucial to understand the properties of normal distribution, including the central limit theorem and how it applies to sampling distributions to solve such problems effectively.