Final answer:
The question pertains to finding probabilities associated with sums of a normally distributed variable related to cholesterol levels. It involves calculating mean sums, standard deviation of sums, z-scores and consulting a standard normal distribution table to determine the requested probabilities and percentages.
Step-by-step explanation:
The question revolves around the concept of probability and the normal distribution, which are fundamental topics in statistics, a branch of mathematics. We are given a normal distribution with a mean (μ) of 180 mg/dL and a standard deviation (σ) of 20 mg/dL for cholesterol levels, while a sample size (n) of 40 is drawn randomly. We'll use the properties of a normal distribution to answer the exercises.
- To find the probability that the sum of the 40 values is greater than 7,500, we first calculate the mean of the sum, which is nμ = 40
180 = 7,200. The standard deviation of the sum is σ√n = 20√40. We then calculate the z-score for the sum of 7,500 and use the standard normal distribution table to find the probability.
- Similarly, for the probability that the sum is less than 7,000, we calculate the z-score for 7,000 and refer to the standard normal distribution table.
- The sum that is one standard deviation above the mean of the sums is 7,200 + (20√40).
- The sum that is 1.5 standard deviations below the mean is 7,200 - (1.5
20√40).
- To find the percentage of sums between 1.5 standard deviations below the mean and one standard deviation above the mean, we'd calculate z-scores for these values and find the corresponding probabilities from the standard normal distribution table; the difference gives us the desired percentage.
Note: Since the exact calculations are not provided, please perform the necessary computations using a standard normal distribution table and z-score formulas.