Question

1. The quality control manager at a drive through restaurant wants to analyze the length of time that a car spends in the drive through line waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of 13.
   1. To obtain probabilities regarding a sample mean using the normal distribution, what sample size is required?
   2. The quality control manager wishes to use a new delivery system designed to get the cars through the drive through faster. A random sample of 50 cars results in a sample mean time spent at the window of 56.2 seconds. What is the probability of obtaining a sample mean time of 56.2 seconds? Based on this probability do you think the new system is effective?
2. A machine is used for filling plastic bottles with a soft drink. The machine is known to have a a target mean of 0 liters and a standard deviation of 0.05 liter.
   1. Suppose you choose a bottle at random filled by this drink dispenser. What is the probability that volume of your bottle has less than 1.98 liters
   2. Suppose you choose 45 bottles at random filled by this dispenser. What is the probability that the mean amount in this sample is less than 1.98 liters?
   3. A quality control manager obtains a random sample of 45 bottles. He will shut down the machine if the sample mean of these 45 values is less than 1.98 liters or above 2.02 liters. What is the probability that the quality control manager will shut down the machine even though the machine is correctly calibrated?

Answer 1

(a)

It is not given that time spent at the window is normally distributed. So to apply central limit theorem for sampling distribution of sample mean the sample size should be at least 30.

Answer: At least 30

(b)

Here we have

The z-score for is

The probability of obtaining a sample mean time of 56.2 seconds is

Since this probability is less than 0.05 so it seems that the new system is not effective.