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The lengths of a large stock of titanium rods follow a normal distribution with a mean (μ) of 350 mm and a standard deviation (σ) of 1 mm. What is the percentage of rods (rounded off to the nearest integer) whose lengths lie between 349 mm and 352 mm?
- a)80
- b)84
Correct answer is between '80,84'. Can you explain this answer?
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Most Upvoted Answer
The lengths of a large stock of titanium rods follow a normal distrib...
Given:
- Mean (μ) = 350 mm
- Standard deviation (σ) = 1 mm
To Find:
The percentage of rods whose lengths lie between 349 mm and 352 mm.
Solution:
To find the percentage of rods, we need to calculate the area under the normal distribution curve between 349 mm and 352 mm. Since the lengths of the rods follow a normal distribution, we can use the z-score formula to find the required percentage.
Step 1: Calculate the z-scores:
The formula for the z-score is given by:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
For the lower value, x1 = 349 mm:
z1 = (349 - 350) / 1 = -1
For the upper value, x2 = 352 mm:
z2 = (352 - 350) / 1 = 2
Step 2: Find the corresponding area:
Once we have the z-scores, we can use a standard normal distribution table or a calculator to find the corresponding area under the curve. The area represents the percentage of rods whose lengths fall within the given range.
Using a standard normal distribution table or calculator, we can find the area corresponding to z1 and z2.
- For z = -1, the area is approximately 0.1587 or 15.87% (rounded off to 2 decimal places).
- For z = 2, the area is approximately 0.9772 or 97.72% (rounded off to 2 decimal places).
Step 3: Calculate the required percentage:
To find the percentage of rods whose lengths lie between 349 mm and 352 mm, we subtract the lower area from the upper area.
Percentage = (0.9772 - 0.1587) * 100 = 81.85%
Rounding off to the nearest integer, the percentage is 82%.
Therefore, the correct answer is option A) 82%.
- Mean (μ) = 350 mm
- Standard deviation (σ) = 1 mm
To Find:
The percentage of rods whose lengths lie between 349 mm and 352 mm.
Solution:
To find the percentage of rods, we need to calculate the area under the normal distribution curve between 349 mm and 352 mm. Since the lengths of the rods follow a normal distribution, we can use the z-score formula to find the required percentage.
Step 1: Calculate the z-scores:
The formula for the z-score is given by:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
For the lower value, x1 = 349 mm:
z1 = (349 - 350) / 1 = -1
For the upper value, x2 = 352 mm:
z2 = (352 - 350) / 1 = 2
Step 2: Find the corresponding area:
Once we have the z-scores, we can use a standard normal distribution table or a calculator to find the corresponding area under the curve. The area represents the percentage of rods whose lengths fall within the given range.
Using a standard normal distribution table or calculator, we can find the area corresponding to z1 and z2.
- For z = -1, the area is approximately 0.1587 or 15.87% (rounded off to 2 decimal places).
- For z = 2, the area is approximately 0.9772 or 97.72% (rounded off to 2 decimal places).
Step 3: Calculate the required percentage:
To find the percentage of rods whose lengths lie between 349 mm and 352 mm, we subtract the lower area from the upper area.
Percentage = (0.9772 - 0.1587) * 100 = 81.85%
Rounding off to the nearest integer, the percentage is 82%.
Therefore, the correct answer is option A) 82%.
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Community Answer
The lengths of a large stock of titanium rods follow a normal distrib...
Z1 = x-(μ)/σ = 349–350/1 = –1
Z2 = 352–350/1 = 2
We know that from –1 to 1 → 68% and
–2 to 2→95%
so from –1 to 2 →68 + (95–68)/2 = 81.5 = 82 %
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The lengths of a large stock of titanium rods follow a normal distribution with a mean (μ) of 350 mm and a standard deviation (σ) of 1 mm. What is the percentage of rods (rounded off to the nearest integer) whose lengths lie between 349 mm and 352 mm?a)80b)84Correct answer is between '80,84'. Can you explain this answer?
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