Final answer:
To calculate the probabilities for the average duration of Alzheimer's disease being less than or greater than 7 years, we use the z-score formula.
The z-score is calculated using the sample data provided, and the associated probabilities are found using a z-table, accounting for the fact that what exceeds 7 years is the complement of what is less than 7 years.
Step-by-step explanation:
The duration of Alzheimer's disease from the appearance of symptoms until death is known to range from 3 to 20 years, with an average of 8 years and a standard deviation of 4 years.
When analyzing the medical records of 30 deceased Alzheimer's patients, the sample mean duration of the disease is used to estimate population parameters and calculate the desired probabilities.
To find the probability of the average duration being less than 7 years (Part a), we can use the z-score formula which is z = (X - μ) / (σ / √ n), where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Since the population mean (μ) is 8, the standard deviation (σ) is 4, and the sample size (n) is 30, we can calculate z as follows:
z = (7 - 8) / (4 / √30) = -1.73
This z-score corresponds to a cumulative probability in the z-table, which gives the probability of the average duration being less than 7 years.
Similarly, for Part b, the probability that the average duration exceeds 7 years is 1 minus the cumulative probability found for z = -1.73.