At a local manufacturing plant, employees must complete new machine set ups within 30 Minutes.New machine set-up times can be described by a normal model with a mean of 22 Minutes and a standard deviation of four minutes.The typical worker needs five minutes to Adjust to his or her surroundings before beginning duties.What percent of new machine set Ups are completed within 25 minutes to allow for this?
A)77.3%
B)27.3%
C)22.7%
D)72.7%
E)none of the above

Question

Quizplus · Accepted Answer

Given that the mean setup time is 22 minutes and the standard deviation is 4 minutes, and workers need an additional 5 minutes to adjust, the effective time for setup to be within 25 minutes is 20 minutes (25 minutes - 5 minutes adjustment time). To find the percentage of setups completed within this time, we calculate the Z-score for 20 minutes: $Z = \frac{20 - 22}{4} = -0.5$. Looking up -0.5 on the standard normal distribution table or using a calculator, we find that approximately 30.85% of values lie below -0.5 standard deviations. However, since the mean is 22 minutes, and we are looking at the time up to 20 minutes, we need to consider the percentage of values from the mean to this Z-score. Since 50% of the distribution is below the mean, and 30.85% is below -0.5 standard deviations, approximately 50% - 30.85% = 19.15% of setups are completed in less than 20 minutes. However, this calculation misinterprets the application of the normal distribution in this context. The correct approach involves recognizing that the question asks for the percentage of setups completed within 25 minutes, accounting for the 5-minute adjustment, effectively making the target time 20 minutes from the start of the actual setup work. The correct calculation involves finding the percentage of the distribution within 20 minutes of the start, not subtracting from 50%. The correct answer involves calculating or looking up the cumulative probability up to a Z-score of -0.5, which actually corresponds to about 30.85% of the distribution being below this point. However, this explanation mistakenly calculates the percentage of setups completed in less than 20 minutes by subtracting from 50%, which is not how the normal distribution's cumulative probabilities are used. The correct approach is to find the cumulative probability up to the Z-score corresponding to 25 minutes (including the adjustment time), which would actually show the proportion of setups completed within the desired timeframe, not the incorrect subtraction method described. The correct interpretation of the Z-score and the cumulative probability from the normal distribution would indicate a higher percentage of setups completed within the adjusted timeframe, aligning more closely with the options provided in the question. The actual calculation should consider the cumulative probability up to the Z-score for 25 minutes (accounting for the adjustment time), which would yield a result indicating the proportion of setups completed within 25 minutes, not the incorrect approach initially described. The correct answer, based on the standard normal distribution and the given mean and standard deviation, would indeed reflect a percentage of setups completed within the desired timeframe, aligning with one of the provided options and demonstrating the application of normal distribution principles to this scenario.

At a Local Manufacturing Plant, Employees Must Complete New Machine