Weights of a certain model of fully loades gravel trucks follow a normal distrbution with mean = 6.4 tons and standard deviation = 0.3 ton. Waht is the probability that a fully loaded truck of this model is
a. less than 6 tons?
b. more that 7 tons?
c. between 6 and 7 tons?
Z = (score-mean)/SD
a. Z = (6-6.4)/.3
Calcuate Z score and find its value in the table in the back of your stat text labeled something like "areas under normal distribution" to get the probability.
Use similar processes for b abd c.
To calculate the probabilities, we can use the standard normal distribution, which is a distribution with a mean of 0 and a standard deviation of 1. We can convert the given values to match the standard normal distribution by using the z-score formula:
z = (x - mean) / standard deviation
where:
z: z-score
x: observed value
mean: mean of the distribution
standard deviation: standard deviation of the distribution
a. To find the probability that a fully loaded truck is less than 6 tons, we need to calculate the z-score for 6 tons and then find the corresponding probability from the standard normal distribution.
z1 = (6 - 6.4) / 0.3
Now, it's time to find the probability using a z-table or a statistical software. For this example, we'll use a z-table.
From the z-table, we look up the z-score of -1.333 and find the corresponding probability of 0.0918. In decimal form, the probability is approximately 0.0918, which is equivalent to 9.18%.
So, the probability that a fully loaded truck of this model is less than 6 tons is approximately 9.18%.
b. To find the probability that a fully loaded truck is more than 7 tons, we calculate the z-score for 7 tons and then find the corresponding probability from the standard normal distribution.
z2 = (7 - 6.4) / 0.3
Again, using the z-table or statistical software, we find the z-score of 2.000 and the corresponding probability of 0.9772. In decimal form, the probability is approximately 0.9772, which is equivalent to 97.72%.
So, the probability that a fully loaded truck of this model is more than 7 tons is approximately 97.72%.
c. To find the probability that a fully loaded truck is between 6 and 7 tons, we need to calculate the z-scores for both values and then find the difference between the probabilities.
z1 = (6 - 6.4) / 0.3
z2 = (7 - 6.4) / 0.3
Again, using the z-table or statistical software, we find the z-score of -1.333 and 2.000, respectively, and their corresponding probabilities of 0.0918 and 0.9772.
To find the probability between the two values, we subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score:
Probability between 6 and 7 tons = Probability (7 tons) - Probability (6 tons)
= 0.9772 - 0.0918
= 0.8854
So, the probability that a fully loaded truck of this model is between 6 and 7 tons is approximately 0.8854, which is equivalent to 88.54%.