A factory’s worker productivity is normally distributed. One worker (worker 1) produces an average of 75 units per day with a standard deviation of 20. Another worker (worker 2) produces at an average rate of 65 per day with a standard deviation of 21. What is the probability that during one week (5 working days) worker 1 will produce more than worker 2?
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To find the probability that worker 1 will produce more than worker 2 during one week, we need to compare the total number of units produced by each worker over 5 days.
Let's start by finding the average number of units produced by each worker over the 5-day period. The average number of units produced by worker 1 in a day is 75, so the average number of units produced in 5 days is 75 * 5 = 375. Worker 2, on the other hand, has an average rate of 65 units per day, so the average number of units produced in 5 days is 65 * 5 = 325.
Next, we need to calculate the standard deviation of the total units produced by each worker over 5 days. Since the units produced by each worker on different days are independent, we can simply add the variances. The variance of worker 1 is (20^2) * 5 = 2000, and the variance of worker 2 is (21^2) * 5 = 2205. So, the standard deviation for worker 1 is sqrt(2000) = 44.7, and the standard deviation for worker 2 is sqrt(2205) = 46.96.
Now, we have converted the problem from one about daily production to one about production over a 5-day period. Since both distributions are normally distributed, we can use the properties of normal distribution to find the probability that worker 1 will produce more than worker 2.
We need to find the probability that worker 1 will produce more than worker 2 over 5 days. To do this, we can use the z-score formula:
z = (x - μ) / σ
where x is the value we are interested in (in this case, 375 - 325 = 50), μ is the mean (375 - 325 = 50), and σ is the standard deviation (sqrt(2000 + 2205) = sqrt(4205) ≈ 64.85).
Now, we can calculate the z-score:
z = (50 - 50) / 64.85 ≈ 0
Finally, we can use a standard normal distribution table or a calculator to find the probability associated with this z-score. In this case, the probability that worker 1 will produce more than worker 2 over 5 days is approximately 0.5, or 50%.