Final answer:
Using the Central Limit Theorem, the standard error for the average service time is calculated, and a z-score is determined to find out the probability that the service time will be less than or equal to 1.1 hours. With a probability of approximately 79.7%, the average planned time might be enough, but there's still a 20.3% chance it could take longer.
Step-by-step explanation:
To determine if 1.1 hours per technician is enough to service a unit, we need to consider the average time that a technician spends servicing a unit and the standard deviation of the time spent. The situation describes a sample with an average service time of 1 hour and a standard deviation of 1 hour. As 70 units constitute a simple random sample, we can apply the Central Limit Theorem to approximate the sampling distribution of the sample mean.
If we assume a normal distribution of servicing times, we can calculate the probability that the actual average service time will be less than or equal to 1.1 hours using the concept of standard error (SE) of the mean. The standard error is calculated using the formula SE = σ/sqrt(n), where σ is the population standard deviation and n is the sample size. In this case, SE = 1 / sqrt(70). When we calculate this, we get an SE of approximately 0.12 hours.
Next, we find the z-score for 1.1 hours. Z = (X - μ) / SE, where X is the value we're checking (1.1 hours), μ is the mean (1 hour), and SE is the standard error. The z-score tells us how many standard errors 1.1 hours is away from the mean. Here, Z = (1.1 - 1) / 0.12 = 0.83. We can then look up this z-score in a standard normal distribution table or use a statistical software to find that the probability of the sample mean being less than or equal to 1.1 hours is about 79.7%.
Will 1.1 hours per technician be enough?
Given that there's roughly 79.7% chance that the average service time for a unit will be less than or equal to 1.1 hours, it suggests that the planned average time would usually be sufficient. However, there is still a 20.3% chance that it will exceed 1.1 hours per unit on average. Depending on the level of risk the company is willing to accept, 1.1 hours may be seen as just adequate, or the company may choose to allow a bit more time to reduce the risk of overruns.