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Probability in statistics is the measure of how likely an event is to occur, given certain conditions. In this exercise, probability is used to evaluate how likely the mean price of gasoline at a sample of service stations is close to the known population mean, which is $2.34.
To calculate probability, we often use the concept of a Z-score, which helps standardize the differences between sample means and population means, making them easier to interpret using a standard normal distribution.
When working with probabilities, it is essential to understand not only how to compute them but also what they mean in a practical context. A higher probability indicates a greater likelihood of observing an event within a certain range, making statistical predictions more reliable.

A sampling distribution refers to the probability distribution of a statistic (like the sample mean) obtained through repeated sampling from a population. In this exercise, we focus on how often we expect the sample mean to fall within a specific range of the population mean when we draw samples of different sizes.
Think of it as gathering a bunch of samples and noting which average happens most frequently. The larger the sample size, the more the sampling distribution of the mean tends to resemble the normal distribution due to the Central Limit Theorem. This effect is crucial in this exercise as we consider sample sizes of 30, 50, and 100.
A well-behaved sampling distribution helps in predicting the likely range of values for the sample mean, given we know the population mean and standard deviation. The shape and spread of the distribution inform us about the likelihood of the sample mean being close to the known population mean.

Standard error (SE) is a key concept in statistics, as it measures how much the sample mean of a population is expected to vary. It gives us an idea of the precision of the sample mean as an estimate of the population mean.
The formula for standard error is given by: \[ SE = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
As you can see, the standard error decreases as the sample size increases. This is evident in the exercise where the SE was calculated to be 0.0365, 0.0283, and 0.020 for sample sizes of 30, 50, and 100, respectively. A smaller standard error suggests that the sample mean is a more accurate reflection of the population mean, making our probability calculations more reliable and the predictions more precise.

The population mean, denoted by \( \mu \), is the average of all the individuals in a population. In this exercise, the population mean is given as $2.34 for the price of gasoline per gallon. This value serves as the benchmark for comparison with our sample means.
Understanding the population mean is essential because it helps in setting the frame of reference for any sample mean derived from the population. It tells us what we expect on average when repeatedly sampling from the population.
In practical scenarios, knowing the population mean enables businesses to make informed decisions. For example, in pricing strategies, understanding the mean gas price helps companies stay competitive. It offers a baseline to gauge deviations from typical conditions in the sampled data and ensure processes are aligned with market realities.