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The salary of workers of a factory is known to follow normal distribution with an average salary of Rs. 10,000 and standard deviation of salary as Rs. 2,000. If 50 workers receive salary more than Rs. 14,000, then the total no. of workers in the factory is
- a)2,193
- b)2,000
- c)2,200
- d)2,500
Correct answer is option 'A'. Can you explain this answer?
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The salary of workers of a factory is known to follow normal distribut...
Normal Distribution and its Properties
Normal distribution is a probability distribution that is symmetric around the mean of the distribution. It is characterized by two parameters - mean (μ) and standard deviation (σ). The probability density function of normal distribution is given by:
f(x) = (1/√(2πσ^2)) * e^(-(x-μ)^2/(2σ^2))
where e is the base of the natural logarithm, π is the mathematical constant pi, and σ^2 is the variance of the distribution.
The properties of normal distribution are:
- It is a continuous distribution that takes on all real values.
- The mean, median, and mode of normal distribution are equal.
- The total area under the curve of normal distribution is equal to 1.
Solution
Given: The mean salary (μ) = Rs. 10,000, the standard deviation of salary (σ) = Rs. 2,000, and the number of workers receiving salary more than Rs. 14,000 = 50.
We need to find the total number of workers in the factory.
Step 1: Find the z-score
The z-score is a measure of how many standard deviations an observation is above or below the mean of the distribution. It is given by:
z = (x - μ)/σ
where x is the observation, μ is the mean, and σ is the standard deviation.
Let x = Rs. 14,000. Then,
z = (14,000 - 10,000)/2,000 = 2
This means that the salary of Rs. 14,000 is 2 standard deviations above the mean.
Step 2: Find the area under the curve
We need to find the area under the curve to the right of z = 2, which represents the proportion of workers receiving salary more than Rs. 14,000. We can use a standard normal distribution table or a calculator to find this area.
Using a standard normal distribution table, we find that the area to the right of z = 2 is 0.0228.
Step 3: Calculate the total number of workers
Let N be the total number of workers in the factory. Then, the number of workers receiving salary more than Rs. 14,000 is given by:
50 = N * 0.0228
Solving for N, we get:
N = 50/0.0228 = 2192.98
Rounding off to the nearest integer, we get:
N = 2193
Therefore, the total number of workers in the factory is 2,193 (option A).
Normal distribution is a probability distribution that is symmetric around the mean of the distribution. It is characterized by two parameters - mean (μ) and standard deviation (σ). The probability density function of normal distribution is given by:
f(x) = (1/√(2πσ^2)) * e^(-(x-μ)^2/(2σ^2))
where e is the base of the natural logarithm, π is the mathematical constant pi, and σ^2 is the variance of the distribution.
The properties of normal distribution are:
- It is a continuous distribution that takes on all real values.
- The mean, median, and mode of normal distribution are equal.
- The total area under the curve of normal distribution is equal to 1.
Solution
Given: The mean salary (μ) = Rs. 10,000, the standard deviation of salary (σ) = Rs. 2,000, and the number of workers receiving salary more than Rs. 14,000 = 50.
We need to find the total number of workers in the factory.
Step 1: Find the z-score
The z-score is a measure of how many standard deviations an observation is above or below the mean of the distribution. It is given by:
z = (x - μ)/σ
where x is the observation, μ is the mean, and σ is the standard deviation.
Let x = Rs. 14,000. Then,
z = (14,000 - 10,000)/2,000 = 2
This means that the salary of Rs. 14,000 is 2 standard deviations above the mean.
Step 2: Find the area under the curve
We need to find the area under the curve to the right of z = 2, which represents the proportion of workers receiving salary more than Rs. 14,000. We can use a standard normal distribution table or a calculator to find this area.
Using a standard normal distribution table, we find that the area to the right of z = 2 is 0.0228.
Step 3: Calculate the total number of workers
Let N be the total number of workers in the factory. Then, the number of workers receiving salary more than Rs. 14,000 is given by:
50 = N * 0.0228
Solving for N, we get:
N = 50/0.0228 = 2192.98
Rounding off to the nearest integer, we get:
N = 2193
Therefore, the total number of workers in the factory is 2,193 (option A).
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Community Answer
The salary of workers of a factory is known to follow normal distribut...
To solve this problem, we need to use the properties of the normal distribution.
Given:
Mean (μ) = Rs. 10,000
Standard Deviation (σ) = Rs. 2,000
Number of workers with salary > Rs. 14,000 = 50
We know that the area under the normal distribution curve represents probabilities. In this case, we need to find the number of workers who receive a salary greater than Rs. 14,000, which corresponds to an area under the curve.
First, we need to calculate the z-score for a salary of Rs. 14,000. The z-score is calculated using the formula:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
z = (14,000 - 10,000) / 2,000
z = 4,000 / 2,000
z = 2
Using the z-score table or a calculator, we can find the area to the right of the z-score of 2. This area represents the proportion of workers with a salary greater than Rs. 14,000.
Looking up the z-score of 2 in the table, we find that the area to the right is approximately 0.0228. This means that approximately 0.0228 of the workers have a salary greater than Rs. 14,000.
To find the total number of workers in the factory, we can set up the following proportion:
(0.0228 / 1) = (50 / x)
Cross-multiplying and solving for x, we get:
0.0228 * x = 50
x = 50 / 0.0228
x ≈ 2192.982
Since we cannot have a fraction of a worker, the closest whole number of workers is 2,193. Therefore, the correct answer is option 'A' - 2,193.
Given:
Mean (μ) = Rs. 10,000
Standard Deviation (σ) = Rs. 2,000
Number of workers with salary > Rs. 14,000 = 50
We know that the area under the normal distribution curve represents probabilities. In this case, we need to find the number of workers who receive a salary greater than Rs. 14,000, which corresponds to an area under the curve.
First, we need to calculate the z-score for a salary of Rs. 14,000. The z-score is calculated using the formula:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
z = (14,000 - 10,000) / 2,000
z = 4,000 / 2,000
z = 2
Using the z-score table or a calculator, we can find the area to the right of the z-score of 2. This area represents the proportion of workers with a salary greater than Rs. 14,000.
Looking up the z-score of 2 in the table, we find that the area to the right is approximately 0.0228. This means that approximately 0.0228 of the workers have a salary greater than Rs. 14,000.
To find the total number of workers in the factory, we can set up the following proportion:
(0.0228 / 1) = (50 / x)
Cross-multiplying and solving for x, we get:
0.0228 * x = 50
x = 50 / 0.0228
x ≈ 2192.982
Since we cannot have a fraction of a worker, the closest whole number of workers is 2,193. Therefore, the correct answer is option 'A' - 2,193.
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The salary of workers of a factory is known to follow normal distribution with an average salary of Rs. 10,000 and standard deviation of salary as Rs. 2,000. If 50 workers receive salary more than Rs. 14,000, then the total no. of workers in the factory isa)2,193b)2,000c)2,200d)2,500Correct answer is option 'A'. Can you explain this answer?
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