The manager of a bakery is analyzing the weight of bread loaves produced per hour during weekdays. The weight of bread loaves follows a Normal distribution with a mean of 500 grams and a standard deviation of 20 grams. (e) Find the total weight of bread loaves in a 4-hour period Y such that on 10% of all 4-hour periods, the total weight is less than this weight.
Question
The manager of a bakery is analyzing the weight of bread loaves produced per hour during weekdays. The weight of bread loaves follows a Normal distribution with a mean of 500 grams and a standard deviation of 20 grams. (e) Find the total weight of bread loaves in a 4-hour period Y such that on 10% of all 4-hour periods, the total weight is less than this weight.
Solution
To solve this problem, we need to use the properties of the normal distribution. Here are the steps:
Step 1: Identify the parameters of the normal distribution. The mean weight of bread loaves produced per hour is 500 grams and the standard deviation is 20 grams.
Step 2: Calculate the mean and standard deviation for a 4-hour period. Since the weight of bread loaves produced per hour follows a normal distribution, the total weight of bread loaves in a 4-hour period also follows a normal distribution. The mean of this distribution is 4 times the mean weight per hour, which is 4*500 = 2000 grams. The standard deviation of this distribution is the square root of 4 times the variance per hour, which is sqrt(4)*20 = 40 grams.
Step 3: Find the z-score corresponding to the 10th percentile. We want to find the total weight Y such that 10% of all 4-hour periods have a total weight less than Y. This corresponds to the 10th percentile of the normal distribution. The z-score corresponding to the 10th percentile is approximately -1.28 (you can find this value in a standard normal distribution table or using a calculator).
Step 4: Convert the z-score to a weight. The formula to convert a z-score to a value in the original distribution is X = μ + Zσ, where X is the value in the original distribution, μ is the mean, Z is the z-score, and σ is the standard deviation. Plugging in the values we have, we get Y = 2000 + (-1.28)*40 = 2000 - 51.2 = 1948.8 grams.
So, on 10% of all 4-hour periods, the total weight of bread loaves is less than 1948.8 grams.
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