Final answer:
To find the probability that the sum of the finishing times for a sample of size 46 is between 6643 and 6676 minutes, we can use the Central Limit Theorem. We can calculate the mean and standard deviation of the sample sum, and then use the z-score formula to find the probability.
Step-by-step explanation:
To find the probability that the sum of the finishing times for a sample of size 46 is between 6643 and 6676 minutes, we can use the Central Limit Theorem. The Central Limit Theorem states that the sum or average of a large enough sample will be approximately normally distributed, regardless of the shape of the original population distribution.
First, we need to calculate the mean and standard deviation of the sample sum. The mean of the sample sum is equal to the mean of the population times the sample size, which is 144 * 46 = 6624 minutes. The standard deviation of the sample sum is equal to the standard deviation of the population times the square root of the sample size, which is 12 * sqrt(46) = 83.792 minutes.
Now, we can find the probability that the sum is between 6643 and 6676 minutes using the z-score formula. The z-score is calculated as (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation. In this case, x1 = 6643, x2 = 6676, μ = 6624, and σ = 83.792. We can then use a z-table or a calculator to find the probability between the two z-scores.