At a large business, employees must report to work at 7:30 A.M. The arrival time

Question

At a large business, employees must report to work at 7:30 A.M. The arrival times of employees is approximately symmetric and mound-shaped with mean 7:22 A.M. and standard deviation 4 minutes. Question 1. Use the 68-95-99.7 rule (also known as the Empirical rule) to determine what percent of employees are late on a typical day. Question 2. A psychological study determined that the typical worker needs four minutes to adjust to their surroundings before beginning their duties. Use the 68-95-99.7 rule (also known as the Empirical rule) to determine what percent of this business employees arrive early enough to make this adjustment before the 7:30 A.M. start time.

Explanation / Answer

NORMAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean time ( u ) = 07.22 AM
standard Deviation ( sd )= 4 Minuetes
68% OF DATA
About 68% of the area under the normal curve is within one standard deviation of the mean. i.e. (u ± 1s.d)
So to the given normal distribution about 68% of the observations lie in between
= [7.22 ± 4]
= [ 7.22 - 4 , 7.22 + 4]
= [ 7.18 AM , 7.26 AM ]
95% OF DATA
About 95% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 2s.d)
So to the given normal distribution about 95% of the observations lie in between
= [7.22 ± 2 * 4]
= [ 7.22 - 2 * 4 , 7.22 + 2* 4]
= [ 7.30 AM , 7.14 AM ]
99.7% OF DATA
About 99.7% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 3s.d)
So to the given normal distribution about 99.7% of the observations lie in between
= [7.22 ± 3 * 4]
= [ 7.22 - 3 * 4 , 7.22 + 3* 4]
= [ 7.34 AM, 7.10 AM ]

PART 1.
95% of data covers below 07:30 AM, and the left over is 5%, and the number of employees
must report work above 07:30 is 5%

PART 2.
if we adjust the time to next 4 minutes, employees make their adjustment time is 07:34. The percentage
is of 99.7% of the area covers under normal curve, we have only 0.30% who covers by that time

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