Final answer:
The probability of a randomly-chosen gas station's gasoline price being more than $0.12 away from the mean is calculated using the normal distribution and associated Z-scores. For average gasoline prices from samples, the Central Limit Theorem dictates that the distribution of sample means will also be normal, with the standard deviation adjusted for sample size. The larger the sample, the smaller the chance of a significant deviation from the population mean.
Step-by-step explanation:
The question at hand involves the application of probability theory and statistics to understand the distribution of gasoline prices. Given a normal distribution with a mean of $2.90 and a standard deviation of $0.40, the likelihood of selecting a gas station with the gasoline price being more than $0.12 away from the mean is found by calculating the area under the normal curve outside of the range $2.78 to $3.02. We first standardize this to a Z-score and then use a Z-table or a statistical calculator to find the corresponding probabilities.
When we are looking at the average price for a sample of gas stations, the distribution of sample means is also normally distributed (due to the Central Limit Theorem), but with a standard deviation equal to the population standard deviation divided by the square root of the sample size (the standard error). This will affect the probabilities in part b, as the standard deviation becomes smaller as the sample size increases, making it less likely for the average price to be more than $0.12 away from the population mean.