Suppose cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12:00 noon and 1:00 pm. A random sample of 40 one-hour periods between 12:00 noon and 1:00 pm is selected and has 22.1 as the sample mean number of cars arriving. (a) Why is the sampling distribution of the sample mean approximately normal? (b) Assuming and , what is the mean and standard deviation of the sampling distribution? What is the probability that a simple random sample of size n = 40 one-hour time periods results in a mean of at least 22.1 cars?
(a) The sampling distribution of the sample mean is approximately normal due to the Central Limit Theorem. This theorem states that for a large enough sample size, the distribution of sample means will approach a normal distribution regardless of the shape of the population distribution.
In this scenario, we have a sample size of 40, which is considered sufficiently large. Therefore, we can assume that the sampling distribution of the sample mean will be approximately normal.
(b) The mean of the sampling distribution is equal to the population mean, which is given as 20 cars per hour.
The standard deviation of the sampling distribution, also known as the standard error of the mean, can be calculated using the formula:
Standard deviation of the sampling distribution = Population standard deviation / Square root of sample size
However, the population standard deviation is not given in the problem statement. Therefore, we cannot determine the exact value of the standard deviation of the sampling distribution.
To find the probability that a simple random sample of size n = 40 one-hour time periods results in a mean of at least 22.1 cars, we need to calculate the z-score and then find the corresponding probability from the standard normal distribution table.
The formula for calculating the z-score is:
z = (sample mean - population mean) / (standard deviation / square root of sample size)
Assuming that the population standard deviation is known or can be estimated, you can plug in the values to calculate the z-score and then find the corresponding probability from the standard normal distribution table or by using a statistical software.