Suppose that cars arrive at Burger King's drive-through at...

Suppose that cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12: 00 noon and 1: 00 P.M. A random sample of 40 one-hour time periods between 12: 00 noon and 1: 00 p.m. is selected and has 22.1 as the mean number of cars arriving. (a) Why is the sampling distribution of $\bar{x}$ approximately normal? (b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$ assuming that $\mu=20$ and $\sigma=\sqrt{20} ?$ (c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this result unusual? What might we conclude?

Suppose that cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12: 00 noon and 1: 00 P.M. A random sample of 40 one-hour time periods between 12: 00 noon and 1: 00 p.m. is selected and has 22.1 as the mean number of cars arriving.
(a) Why is the sampling distribution of $\bar{x}$ approximately normal?
(b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$ assuming that $\mu=20$ and $\sigma=\sqrt{20} ?$
(c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this result unusual? What might we conclude?

Key Concepts

Unusual Event and Conclusion

When the computed probability of observing a sample mean as extreme as the one obtained is very low, it is considered statistically unusual. This low probability may indicate that the observed sample mean deviates significantly from what is expected under the assumed population parameters, potentially prompting further investigation or suggesting that the underlying assumptions about the population may need to be reconsidered.

Standard Normal Distribution and Z-score

The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one. Using the Z-score, which standardizes a value by measuring how many standard errors it is away from the population mean, allows one to determine probabilities and assess how unusual a particular sample mean is. This transformation enables the application of standard normal tables or computational tools to calculate tail probabilities.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of the means of many samples, each of a given size drawn from the same population. It has a mean equal to the population mean and a spread (variability) measured by the standard error, which decreases as the sample size increases. This concept allows statisticians to understand and quantify the variation expected in sample means due to random sampling.

Central Limit Theorem

The central limit theorem is a fundamental concept in probability that states that, regardless of the population's original distribution, the distribution of the sample means will tend to be normal (or approximately normal) as the sample size becomes large. This is critical when dealing with sample means because it justifies using normal probability methods when the sample size is sufficiently large—even if the individual observations are not normally distributed.

Standard Error

The standard error is a measure of the variability or dispersion of the sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size. The standard error quantifies how much the sample mean is expected to fluctuate from sample to sample and is crucial for constructing confidence intervals and conducting hypothesis tests.

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