STATISTICS PLEASE HELP!!!!! I am sorry this is jy second time posting, but I really need help. I have tried contacting my teacher and classmates with no luck. Please just steer me in the right direction!:

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? Is this result unusual?

(a) The sampling distribution of the sample mean is approximately normal due to the Central Limit Theorem (CLT). According to the CLT, when the sample size is large enough (typically n > 30), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

(b) The mean of the sampling distribution will be the same as the population mean, which is 20 cars per hour. The standard deviation of the sampling distribution (also known as the standard error) can be calculated using the formula:

Standard deviation (σ) = Population standard deviation / Square root of sample size

Since the population standard deviation is not given in the question, we cannot calculate the exact value of the standard deviation.

To find the probability that a random sample of size n = 40 one-hour time periods results in a mean of at least 22.1 cars, we need to standardize the value 22.1 by subtracting the population mean (20) and dividing by the standard deviation. Once we have the standardized value, we can use the normal distribution table or a statistical software to find the corresponding probability.

Without the specific values for standard deviation, we cannot determine whether the result is unusual or not.

To answer this question, we need to understand the concept of sampling distributions and how they relate to the population distribution.

(a) The sampling distribution of the sample mean is approximately normal because of the Central Limit Theorem. This theorem states that if the sample size is large enough, regardless of the shape of the population distribution, the sampling distribution of the sample mean will approach a normal distribution. In this case, the sample size of 40 is reasonably large enough to invoke the Central Limit Theorem.

(b) We are given that the population mean arrival rate is 20 cars per hour, and the population standard deviation is not specified. However, we can calculate the mean and standard deviation of the sampling distribution using the population mean and the standard error formula.

The mean of the sampling distribution is equal to the population mean. In this case, the mean of the sampling distribution is 20 cars per hour.

The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:

Standard Error = Population Standard Deviation / √(Sample Size)

Since we are not given the population standard deviation, we cannot calculate the exact standard error. However, assuming we know it is 5 cars per hour, we can calculate the standard error using the formula. For example:

Standard Error = 5 / √40 = 0.7906

To find the probability that a simple random sample of size 40 one-hour time periods results in a mean of at least 22.1 cars, we can standardize the sample mean using the sampling distribution. We can then calculate the z-score and find the probability using a standard normal distribution table or a statistical calculator.

Z-score = (Sample Mean - Population Mean) / Standard Error
Z-score = (22.1 - 20) / 0.7906 = 2.931

Using a standard normal distribution table or a statistical calculator, we can find the probability that a z-score is greater than or equal to 2.931. Assuming a two-tailed test, the probability is the area to the right of the z-score, which is approximately 0.0017 or 0.17%.

This result is considered unusual because it has a very low probability of occurring by chance alone.