Final answer:
To calculate the probability that the sum of the 50 values is more than 2,400, apply the Central Limit Theorem. Calculate the mean and standard deviation of the sum, convert the given sum to a z-score, and find the probability corresponding to that z-score. The result will show the likelihood of the sum of the sample exceeding 2,400.
Step-by-step explanation:
To find the probability that the sum of the 50 values is more than 2,400, we need to apply the Central Limit Theorem. Since we are looking at the sum of the samples, rather than the sample mean, we will first need to calculate the mean of the sum and the standard deviation of the sum.
The mean of the sample sum, μ_sum, is the sample size, n, multiplied by the population mean, μ: μ_sum = n * μ = 50 * 45 = 2250. The standard deviation of the sample sum, σ_sum, is the square root of the sample size, √n, multiplied by the population standard deviation, σ: σ_sum = √n * σ = √50 * 8 = √400 * 8 = 20 * 8 = 160.
Next, we convert the sum we are interested in, which is 2,400, to a z-score. The z-score is given by (X - μ_sum) / σ_sum = (2400 - 2250) / 160 = 0.9375. We then use a standard normal distribution table or a calculator with normal distribution functions to find the probability of a z-score being more than 0.9375. Let's call this value P.
Interpretation: The value P represents the probability that the sum of the 50 randomly drawn values from a distribution with a mean of 45 and a standard deviation of 8 will be greater than 2,400. If P is small, it suggests that such a sum is unlikely; if P is large, it suggests the sum is more common.