Question
A worker at a landscape design center uses a machine to fill bags with potting soil. Assume that the quantity put in each bag follows the continuous uniform distribution with low and high filling weights of 7.5 pounds and 11.6 pounds, respectively.
a. Calculate the expected value and the standard deviation of this distribution. (Do not round intermediate calculations. Round your "Expected value" to 2 decimal places and "Standard deviation" answer to 4 decimal places.)
b. Find the probability that the weight of a randomly selected bag is no more than 8.8 pounds. (Round your answer to 2 decimal places.)
c. Find the probability that the weight of a randomly selected bag is at least 8.3 pounds. (Round your answer to 2 decimal places.)
a. Calculate the expected value and the standard deviation of this distribution. (Do not round intermediate calculations. Round your "Expected value" to 2 decimal places and "Standard deviation" answer to 4 decimal places.)
b. Find the probability that the weight of a randomly selected bag is no more than 8.8 pounds. (Round your answer to 2 decimal places.)
c. Find the probability that the weight of a randomly selected bag is at least 8.3 pounds. (Round your answer to 2 decimal places.)
Answers
GPT 3.5
a. The expected value of a continuous uniform distribution can be calculated as the average of the low and high filling weights:
Expected value = (low + high) / 2 = (7.5 + 11.6) / 2 = 9.55 pounds
The standard deviation of a continuous uniform distribution can be calculated using the formula:
Standard deviation = (high - low) / sqrt(12)
Standard deviation = (11.6 - 7.5) / sqrt(12) ≈ 1.4285 pounds
b. To find the probability that the weight of a randomly selected bag is no more than 8.8 pounds, we need to calculate the area of the probability density function (PDF) below 8.8 pounds.
The probability density function of a continuous uniform distribution is a constant value between the low and high filling weights. Since the range of the distribution is 11.6 - 7.5 = 4.1 pounds, the PDF is 1/4.1 within this range.
To calculate the probability, we need to find the area under the PDF curve between 7.5 and 8.8 pounds:
Probability = (8.8 - 7.5) / 4.1 ≈ 0.3171
Therefore, the probability that the weight of a randomly selected bag is no more than 8.8 pounds is approximately 0.32.
c. To find the probability that the weight of a randomly selected bag is at least 8.3 pounds, we need to calculate the area of the probability density function (PDF) above 8.3 pounds.
The PDF is still 1/4.1 within the range of 7.5 and 11.6 pounds.
To calculate the probability, we need to find the area under the PDF curve above 8.3 pounds:
Probability = (11.6 - 8.3) / 4.1 ≈ 0.8049
Therefore, the probability that the weight of a randomly selected bag is at least 8.3 pounds is approximately 0.80.
Expected value = (low + high) / 2 = (7.5 + 11.6) / 2 = 9.55 pounds
The standard deviation of a continuous uniform distribution can be calculated using the formula:
Standard deviation = (high - low) / sqrt(12)
Standard deviation = (11.6 - 7.5) / sqrt(12) ≈ 1.4285 pounds
b. To find the probability that the weight of a randomly selected bag is no more than 8.8 pounds, we need to calculate the area of the probability density function (PDF) below 8.8 pounds.
The probability density function of a continuous uniform distribution is a constant value between the low and high filling weights. Since the range of the distribution is 11.6 - 7.5 = 4.1 pounds, the PDF is 1/4.1 within this range.
To calculate the probability, we need to find the area under the PDF curve between 7.5 and 8.8 pounds:
Probability = (8.8 - 7.5) / 4.1 ≈ 0.3171
Therefore, the probability that the weight of a randomly selected bag is no more than 8.8 pounds is approximately 0.32.
c. To find the probability that the weight of a randomly selected bag is at least 8.3 pounds, we need to calculate the area of the probability density function (PDF) above 8.3 pounds.
The PDF is still 1/4.1 within the range of 7.5 and 11.6 pounds.
To calculate the probability, we need to find the area under the PDF curve above 8.3 pounds:
Probability = (11.6 - 8.3) / 4.1 ≈ 0.8049
Therefore, the probability that the weight of a randomly selected bag is at least 8.3 pounds is approximately 0.80.