Answer :
a. To sketch the distribution of breaking points for the rope, we can create a normal distribution curve using the mean and standard deviation provided.
The mean breaking point is 5036 lbs, and the standard deviation is 122 lbs. The curve should be centered around the mean with a spread determined by the standard deviation.
b. To determine the proportion of ropes that will break with a 5000 lbs load, we need to calculate the z-score and then find the corresponding area under the normal distribution curve. The z-score can be calculated using the formula:
z = (x - μ) / σ
where x is the load value, μ is the mean breaking point, and σ is the standard deviation. In this case, x = 5000 lbs, μ = 5036 lbs, and σ = 122 lbs.
Substituting the values:
z = (5000 - 5036) / 122
z = -0.295
Using a standard normal distribution table or a calculator, we can find the proportion or area under the curve corresponding to a z-score of -0.295. This will give us the proportion of ropes that will break with a load of 5000 lbs.
c. To determine the load at which 95% of all ropes will break, we need to find the corresponding z-score that encloses an area of 0.95 under the normal distribution curve. We can then use the z-score to calculate the load value using the formula:
x = μ + z * σ
where x is the load value, μ is the mean breaking point, σ is the standard deviation, and z is the z-score.
Using a standard normal distribution table or a calculator, we can find the z-score that encloses an area of 0.95. Substituting the z-score into the formula, we can calculate the load value at which 95% of all ropes will break.
Please note that in this case, we assume the breaking points of the rope are normally distributed and use the properties of the normal distribution to make these calculations.
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At a load of approximately 5193 lbs, 95% of all ropes will break.
a. The distribution of breaking points for this rope would be a normal distribution with a mean of 5036 lbs and a standard deviation of 122 lbs.
b. To find the proportion of ropes that will break with 5000 lbs of load, we need to standardize the value using the formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean breaking point, and σ is the standard deviation.
z = (5000 - 5036) / 122
z = -0.295
Using a standard normal distribution table or calculator, we can find that the proportion of ropes that will break with 5000 lbs of load is approximately 0.385 (or 38.5%).
c. To find the load at which 95% of all ropes will break, we need to find the z-score that corresponds to the 95th percentile of the normal distribution. We can use a standard normal distribution table or calculator to find this value, which is approximately 1.645.
Then, we can use the formula z = (x - μ) / σ and solve for x:
1.645 = (x - 5036) / 122
x = 5193.29
Therefore, at a load of approximately 5193 lbs, 95% of all ropes will break.
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