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Answer :
The probability that the next 15 wheel throwing processes will take a combined time between 12 and 13 hours is approximately 0.8141.
We can use the Central Limit Theorem to solve this problem. The sum of normally distributed random variables is also normally distributed. Therefore, the total time taken for the next 15 wheel throwing processes will be normally distributed with a mean of 15 multiplied by the mean of each individual process, and a standard deviation of √(15 multiplied by the variance of each individual process).
Mean of each individual process = 50 minutes
Variance of each individual process = [tex](standard deviation)^2[/tex]
= [tex]6^2[/tex]
= 36 [tex]minutes^2[/tex]
Mean of the total time taken = 15 multiplied by 50
= 750 minutes
Standard deviation of the total time taken = √(15 multiplied by 36)
= √(540) ≈ 23.24 minutes
To convert the hours to minutes, 12 hours is equal to 12 multiplied by 60 = 720 minutes, and 13 hours is equal to 13 multiplied by 60 = 780 minutes.
Thus, we need to find the probability that the total time taken is between 720 minutes and 780 minutes. We can standardize the values using the z-score formula:
z = (x - mean) / standard deviation
Using this formula, we can calculate the z-scores for 720 minutes and 780 minutes using the mean and standard deviation calculated above.
z-score for 720 minutes = (720 - 750) / 23.24 ≈ -1.29
z-score for 780 minutes = (780 - 750) / 23.24 ≈ 1.29
Next, we can use the standard normal distribution table or a calculator to find the probability that the z-score falls between -1.29 and 1.29. This probability represents the probability that the total time taken for the next 15 wheel throwing processes will be between 12 hours and 13 hours.
By looking up the z-scores in the standard normal distribution table or using a calculator, we find that the probability is approximately 0.8141.
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