A trucking firm delivers appliances for a large retail operation. The packages (or crates) have a mean weight of 306 lb. and a variance of 2209. (Give your answers correct to four decimal places.)
(a) If a truck can carry 3940 lb. and 25 appliances need to be picked up, what is the probability that the 25 appliances will have an aggregate weight greater than the truck's capacity? Assume that the 25 appliances represent a random sample.
(b) If the truck has a capacity of 7810 lb., what is the probability that it will be able to carry the entire lot of 25 appliances?
To solve these probability questions, we need to use the information provided about the mean weight and variance of the packages. We'll be working with the normal distribution because the sample size is large enough.
(a) To find the probability that the aggregate weight of 25 appliances will be greater than the truck's capacity of 3940 lb., we need to calculate the probability of the sum being greater than 3940. Let's denote X as the weight of a single appliance.
The mean weight of a single appliance is given as 306 lb. with a variance of 2209. Hence, the standard deviation (σ) can be found by taking the square root of the variance:
σ = √(2209) = 47.
Now, we can use the Central Limit Theorem and assume that the sum of 25 appliances will be approximately normally distributed with a mean of 25 * 306 = 7650 lb. and a standard deviation of √(25 * 2209) = 235.0579127 lb.
To calculate the probability, we'll convert the problem into the standard normal distribution by using the z-score formula:
z = (x - mean) / standard deviation,
where x is the value for which we want to find the probability.
In this case, we want to find the probability that the sum of 25 appliances is greater than 3940 lb. We substitute these values into the formula:
z = (3940 - 7650) / 235.0579127,
z = -11.4159.
To find the probability, we use a standard normal distribution table or calculator to look up the value for the z-score of -11.4159. This probability will be the percentage of the area under the curve to the right of -11.4159.
(b) To find the probability that the truck will be able to carry the entire lot of 25 appliances with a capacity of 7810 lb., we need to calculate the probability that the sum is less than or equal to 7810 lb.
Using the same steps as above, we calculate the z-score:
z = (7810 - 7650) / 235.0579127,
z = 0.6811.
Again, we use a standard normal distribution table or calculator to find the probability corresponding to a z-score of 0.6811. This probability will be the percentage of the area under the curve to the left of 0.6811.
By following these steps, you can find the probabilities for both parts (a) and (b) of the question.