A particular automobile costs an average of 21,755 in the...

A particular automobile costs an average of $\$ 21,755$ in the Pacific Northwest. The standard deviation of prices is $\$ 650 .$ Suppose a random sample of 40 dealerships in Washington and Oregon is taken and their managers are asked what they charge for this automobile. What is the probability of getting a sample average cost of less than $\$ 21,500 ?$ Assume that only 120 dealerships in the entire Pacific Northwest sell this automobile.

Key Concepts

Sampling Distribution

This concept refers to the probability distribution of a statistic (such as the sample mean) calculated from a set of samples drawn from the population. In the context of the problem, it involves understanding how the sample average cost varies from samples of 40 dealerships and that this distribution is centered at the true population mean with variability dictated by the standard error.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the original population distribution, the sampling distribution of the mean will approximate a normal distribution provided the sample size is sufficiently large. Here, even if the underlying distribution of automobile prices is unknown or non-normal, the theorem justifies using a normal approximation to calculate the probability related to the sample mean.

Standard Error

The standard error is the measure of the variability of the sample mean from sample to sample, calculated as the population standard deviation divided by the square root of the sample size. It quantifies the precision of the sample mean as an estimate of the population mean and plays a crucial role in determining the likelihood of observing a sample mean below a given threshold.

Finite Population Correction

When the sample size is drawn without replacement from a finite population, the variability of the sample mean is slightly reduced. The finite population correction factor adjusts the standard error by multiplying it by the square root of [(Population Size - Sample Size) / (Population Size - 1)], reflecting the impact of the limited population on the estimate's precision.

Z-Score Transformation

This is the process of standardizing the sample mean by subtracting the population mean and dividing by the (corrected) standard error. The Z-score indicates how many standard errors the sample mean is away from the population mean, allowing one to use the standard normal distribution to compute probabilities, such as the chance of obtaining a sample mean below a specified value.

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