Final answer:
The 95% confidence interval for the mean price of gasoline, based on a sample mean of $4.39 and a standard deviation of $0.23 from 30 gas stations, is calculated to be between $4.31 and $4.47. This interval implies that we can be 95% confident that the actual population mean for gas prices falls within this range.
Step-by-step explanation:
To answer the student question, first, we calculate a 95% confidence interval for the mean price of gasoline from the given sample data. Since the sample size is less than 30, we would typically use the t-distribution; however, since we have 30 data points exactly, it's okay to use the z-distribution as an approximation.
The formula for a confidence interval using the z-distribution is:
CI = µ ± (z * (σ / √n))
Here, μ is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the sample standard deviation, and n is the sample size.
To find the z-score associated with a 95% confidence level, we look at the z-table and find that the z-score is approximately 1.96. Using the sample mean of $4.39, the standard deviation of $0.23, and the sample size of 30, the confidence interval is calculated as follows:
CI = $4.39 ± (1.96 * ($0.23 / √30))
Calculate the margin of error (ME):
ME = 1.96 * ($0.23 / √30) ≈ $0.08
Then, plug in the values to get the confidence interval:
CI = $4.39 ± $0.08
CI = ($4.31, $4.47)
This confidence interval suggests that we can be 95% confident that the population mean price of gasoline will fall between $4.31 and $4.47.