Use the table to answer each question. Note: Round z-scores to the nearest hundredth and then find the required A values using the table.
The breaking point of a particular type of rope is normally distributed, with a mean of 310 pounds and a standard deviation of 24 pounds. What is the probability that a piece of this rope chosen at random will have the following breaking points? (Round your answers to three decimal places.)
(a) less than 280 pounds
(b) between 300 and 330 pounds
Well, well, well, breaking ropes, huh? Let's see what we can do with those z-scores!
(a) So you want to know the probability that a random piece of this rope will have a breaking point less than 280 pounds? To find that out, we need to calculate the z-score first.
To do that, we'll use the formula: z = (X - μ) / σ
Where X is the value we're interested in (in this case 280 pounds), μ is the mean (310 pounds), and σ is the standard deviation (24 pounds).
So, the z-score for a breaking point of 280 pounds is: z = (280 - 310) / 24 = -1.25 (rounded to the nearest hundredth).
Now, let's consult our handy-dandy table and find the corresponding A value for a z-score of -1.25. *rifles through a stack of papers* Ah, there it is!
According to the table, the A value for a z-score of -1.25 is 0.105 (rounded to three decimal places).
Therefore, the probability that a random piece of this rope will have a breaking point less than 280 pounds is 0.105, or in percentage form, 10.5%. It seems like a risky rope!
(b) Now, you want to know the probability that the breaking point of a random piece of this rope falls between 300 and 330 pounds. We'll need to calculate two z-scores for this one.
For the lower bound of 300 pounds, the z-score is: z = (300 - 310) / 24 = -0.42 (rounded to the nearest hundredth).
For the upper bound of 330 pounds, the z-score is: z = (330 - 310) / 24 = 0.83 (rounded to the nearest hundredth).
Now, let's find the corresponding A values for these z-scores. *flips through the pages* Ah, there they are!
The A value for a z-score of -0.42 is 0.335 (rounded to three decimal places), and the A value for a z-score of 0.83 is 0.798 (rounded to three decimal places).
To find the probability between two values, we subtract the A value of the lower bound from the A value of the upper bound. So, 0.798 - 0.335 = 0.463.
Therefore, the probability that a random piece of this rope will have a breaking point between 300 and 330 pounds is 0.463, or 46.3%.
Remember, these probabilities are based on a normal distribution, so they might not fully capture the unpredictable nature of real-life rope-breaking situations. Good luck, and don't hang on too tightly!
To find the probability for each breaking point, we need to convert the given breaking points into z-scores using the formula:
z = (x - μ) / σ
where z is the z-score, x is the breaking point, μ is the mean breaking point, and σ is the standard deviation.
(a) For the breaking point less than 280 pounds:
z = (280 - 310) / 24
z = -1.25
From the z-score table, we can find the probability corresponding to a z-score of -1.25. The closest value in the table is -1.2, which corresponds to a probability of 0.1056.
(b) For the breaking point between 300 and 330 pounds:
To find the probability between two breaking points, we need to find the z-scores for both points and then subtract the cumulative probabilities of the lower breaking point from the cumulative probabilities of the higher breaking point.
z1 = (300 - 310) / 24
z1 = -0.42
z2 = (330 - 310) / 24
z2 = 0.83
From the z-score table, we can find the cumulative probability for each z-score:
P(z < -0.42) = 0.3365
P(z < 0.83) = 0.7967
Now, we can calculate the probability between 300 and 330 pounds:
P(300 < x < 330) = P(z < 0.83) - P(z < -0.42)
P(300 < x < 330) = 0.7967 - 0.3365
P(300 < x < 330) = 0.4602
Therefore, the probability that a piece of this rope chosen at random will have a breaking point less than 280 pounds is 0.106, and the probability that it will have a breaking point between 300 and 330 pounds is 0.460.
To find the probabilities for the breaking points, we need to use the standard normal distribution table.
(a) To find the probability that a piece of rope chosen at random will have a breaking point less than 280 pounds, we need to find the corresponding z-score and look up the value in the standard normal distribution table.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = the value you want to find the probability for (280 pounds)
μ = the mean of the distribution (310 pounds)
σ = the standard deviation of the distribution (24 pounds)
Substituting the values into the z-score formula, we have:
z = (280 - 310) / 24 = -1.25
Now, we need to find the probability associated with a z-score of -1.25 in the standard normal distribution table. Looking up -1.25 in the table, we find that the corresponding area/probability is 0.1056.
Therefore, the probability that a randomly chosen piece of rope will have a breaking point less than 280 pounds is approximately 0.106.
(b) To find the probability that a piece of rope chosen at random will have a breaking point between 300 and 330 pounds, we need to find the probabilities for each endpoint separately and then subtract them.
First, we find the z-score for 300 pounds:
z = (300 - 310) / 24 = -0.42
Looking up -0.42 in the standard normal distribution table, we find that the corresponding area/probability is 0.3365.
Next, we find the z-score for 330 pounds:
z = (330 - 310) / 24 = 0.83
Looking up 0.83 in the standard normal distribution table, we find that the corresponding area/probability is 0.7977.
To find the probability between 300 and 330 pounds, we subtract the probability of 300 pounds from the probability of 330 pounds:
0.7977 - 0.3365 = 0.4612
Therefore, the probability that a randomly chosen piece of rope will have a breaking point between 300 and 330 pounds is approximately 0.461.