Final answer:
To find the probability that the sum of 40 cholesterol test values is greater than 7,550, we apply the Central Limit Theorem to determine the mean and standard deviation of the sum, calculate the z-score, and then use a normal distribution table to find the corresponding probability.
Step-by-step explanation:
The question asks us to find the probability that the sum of 40 random cholesterol test values is greater than 7,550. Given that the population mean is 180 and the standard deviation is 20, we utilize the Central Limit Theorem (CLT) to solve this. CLT states that the sampling distribution of the sum or mean of a large enough sample drawn from a population will be approximately normally distributed, regardless of the population's distribution.
First, we compute the mean of the sum of the 40 values as the sample size (40) multiplied by the population mean (180), which equals 7,200. Then, we calculate the standard deviation of the sum as the population standard deviation (20) multiplied by the square root of the sample size (√40), which equals approximately 126.49.
Next, we standardize the threshold for the sum using the z-score formula:
Z = (X - μSUM) / σSUM
Where X is the threshold sum (7,550), μSUM is the mean sum (7,200), and σSUM is the standard deviation of the sum (126.49). The resulting z-score tells us how many standard deviations the threshold is away from the mean sum. Finally, we use a standard normal distribution table or a calculator with normal distribution functionality to find the probability that corresponds to this z-score.