determine the probability that a sample of 49 boxes will have a mean weight below 365 grams. past records show that the population mean is 368 grams and the population standard deviation is 7. the probability will be between.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply by 49.

To determine the probability that a sample of 49 boxes will have a mean weight below 365 grams, we will use the Central Limit Theorem and the properties of the normal distribution.

The Central Limit Theorem states that, for a large enough sample size, the sample mean will be normally distributed regardless of the shape of the population distribution.

Step 1: Find the standard deviation of the sampling distribution.
Since the population standard deviation (σ) is known and the sample size (n) is 49, we can use the formula for the standard deviation of the sampling distribution (also known as the standard error, SE):
SE = σ / √n

SE = 7 / √49
SE = 7/7
SE = 1

Step 2: Standardize the score.
To calculate the probability, we need to standardize the score by converting it into a z-score. The formula for the z-score is:
z = (x - μ) / SE

where x is the sample mean (365 grams), μ is the population mean (368 grams), and SE is the standard deviation of the sampling distribution (1).

z = (365 - 368) / 1
z = -3 / 1
z = -3

Step 3: Calculate the probability.
We need to find the probability of obtaining a z-score less than or equal to -3. For this, we can consult a standard normal distribution table (also known as a z-table) or use a statistical software.

Using a z-table or calculator, we find that the probability of obtaining a z-score less than or equal to -3 is approximately 0.0013.

Therefore, the probability that a sample of 49 boxes will have a mean weight below 365 grams is approximately 0.0013.