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The monthly income of 5,000 workers at the Microsoft plant are distributed normally. Suppose the mean monthly income is $1,250 and the standard deviation is $250. a) How many workers earn more than $1500 per month? b) How many workers earn less than $750 per month? c) What percentage of the workers earn between $750 and $1500 per month? d) What percentage of the workers earn less than $1750 per month?

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Final answer:

a) Approximately 15.87% of the workers earn more than $1500 per month. b) Approximately 2.28% of the workers earn less than $750 per month. c) Approximately 81.85% of the workers earn between $750 and $1500 per month. d) Approximately 97.72% of the workers earn less than $1750 per month.

Step-by-step explanation:

a) To find the number of workers who earn more than $1500 per month, we need to find the area to the right of the value $1500 on the normal distribution curve. We can use the z-score formula to calculate this. First, we find the z-score of $1500 using the formula: Z = (X - μ) / σ, where X is the value we want to find the area to the right of, μ is the mean, and σ is the standard deviation. In this case, X = $1500, μ = $1250, and σ = $250. Plugging in these values, we get: Z = (1500 - 1250) / 250 = 1. The z-score represents the number of standard deviations a value is from the mean. So, a z-score of 1 tells us that $1500 is 1 standard deviation above the mean. Using a z-table or calculator, we can find that the area to the right of a z-score of 1 is approximately 0.1587. Therefore, approximately 15.87% of the workers earn more than $1500 per month.

b) To find the number of workers who earn less than $750 per month, we need to find the area to the left of the value $750 on the normal distribution curve. We can use the same z-score formula as in part a. Plugging in the values X = $750, μ = $1250, and σ = $250, we get: Z = (750 - 1250) / 250 = -2. The z-score represents the number of standard deviations a value is from the mean. So, a z-score of -2 tells us that $750 is 2 standard deviations below the mean. Using a z-table or calculator, we can find that the area to the left of a z-score of -2 is approximately 0.0228. Therefore, approximately 2.28% of the workers earn less than $750 per month.

c) To find the percentage of workers who earn between $750 and $1500 per month, we need to find the area between the values $750 and $1500 on the normal distribution curve. We can subtract the area to the left of $750 from the area to the left of $1500 to find this. Using the z-score formulas from parts a and b, we can find the areas to the left of $750 and $1500. The area to the left of $750 is approximately 0.0228 (found in part b). The area to the left of $1500 is approximately 0.8413 (found in part a). Subtracting these two values, we get: 0.8413 - 0.0228 = 0.8185. Therefore, approximately 81.85% of the workers earn between $750 and $1500 per month.

d) To find the percentage of workers who earn less than $1750 per month, we need to find the area to the left of the value $1750 on the normal distribution curve. We can use the same z-score formula as in part a. Plugging in the values X = $1750, μ = $1250, and σ = $250, we get: Z = (1750 - 1250) / 250 = 2. The z-score represents the number of standard deviations a value is from the mean. So, a z-score of 2 tells us that $1750 is 2 standard deviations above the mean. Using a z-table or calculator, we can find that the area to the left of a z-score of 2 is approximately 0.9772. Therefore, approximately 97.72% of the workers earn less than $1750 per month.

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