Final answer:
To find the 99.5% confidence interval for mean efficiency of electric motors, given an average efficiency of 85% and standard deviation of 2% for a sample of 60 motors, we use a Z-Score of approximately 2.807. The interval is calculated to be (84.276%, 85.724%).
Step-by-step explanation:
To construct a 99.5% confidence interval for the mean efficiency of electric motors, first, understand that the efficiency of electric motors is quite significant. A more efficient motor will consume less energy and provide more power output. Efficiency is a key performance index, especially in an era of energy conservation and sustainable practices. The average efficiency of a 60-motor sample is given as 85% with a standard deviation of 2%.
To find the confidence interval, it is essential to determine the appropriate Z-Score, which corresponds to the chosen confidence level. For a 99.5% confidence interval, the Z-Score is typically found in a standard normal distribution table, which in this case, is approximately 2.807. The formula used to calculate the confidence interval is then:
CI = ± (Z * (sd/√n))
Where CI is the confidence interval, Z is the Z-Score, sd is the standard deviation, and n is the sample size. Plugging in the values, we get:
CI = ± (2.807 * (2/√60))
Calculating this yields:
CI = ± (2.807 * (2/7.746))
CI = ± (0.724)
Therefore, the 99.5% confidence interval is:
85 ± 0.724 = (84.276, 85.724)
Hence, we are 99.5% confident that the true mean efficiency of electric motors is between 84.276% and 85.724%.