Probability Calculations for Normal Distribution with Sample Size of 100
The image contains four separate parts (a through d) of a question regarding a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 4. It then asks about the probabilities related to a sample (n) of 100. Here are the answers to each of the parts as provided in the image:
a. What is the probability that X is less than 49?
The answer is given as P(X < 49) = 0.0062.
b. What is the probability that X is between 49 and 50.5?
The answer is given as P(49 < X < 50.5) = 0.8822.
c. What is the probability that X is above 50.6?
The answer is given as P(X > 50.6) = 0.0688.
d. There is a 40% chance that X is above what value?
The answer is given as X = 50.1012.
The calculations for these probabilities would typically involve using the standard normal distribution (Z-distribution), where you would standardize the values using the formula \( Z = \frac{X - μ}{σ} \). Since the sample size is 100, the standard error of the mean (SEM) would also be involved, computed by \( SEM = \frac{σ}{\sqrt{n}} \). However, the full workings of these calculations are not shown in the image; it only provides the final numerical results. If you need further assistance on how to perform these calculations, please feel free to ask!
