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 this question, we need to use the properties of the normal distribution.
(a) To find the probability that the aggregate weight of the 25 appliances will be greater than the truck's capacity, we need to calculate the probability of the sample mean of the 25 appliances being greater than the truck's capacity.
The sample mean (M) is equal to the mean of the population (mu) which is 306 lb.
The standard deviation of the population (sigma) is the square root of the variance which is given as 2209 lb.
Now, the standard error (SE) of the sample mean can be calculated using the formula:
SE = sigma / sqrt(n)
where n is the sample size. In this case, n = 25.
SE = 2209 / sqrt(25) = 441.8 lb.
We can use a z-score to find the probability. The z-score is calculated as:
z = (X - M) / SE
where X is the truck's capacity, which is 3940 lb.
z = (3940 - 306) / 441.8
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to this z-score. The probability will be the area under the curve to the right of the z-score.
(b) To find the probability that the truck will be able to carry the entire lot of 25 appliances, we need to calculate the probability that the aggregate weight of the 25 appliances is less than or equal to the truck's capacity.
To do this, we follow the same steps as in part (a), but instead of finding the probability to the right of the z-score, we find the probability to the left of the z-score.
Once we find the corresponding probability using the z-score, we have the answer to part (b).