Final answer:
To calculate the probability, we need to find the z-score for a difference of 0.5 gallons and then use the standard normal distribution to find the area under the curve beyond that z-score. The probability that the sample mean would differ from the population mean by more than 0.5 gallons is 1.3836.
Step-by-step explanation:
To find the probability that the sample mean would differ from the population mean by more than 0.5 gallons, we need to use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means is approximately normally distributed, regardless of the shape of the population distribution, when the sample size is large enough. In this case, since the sample size is 39 (which is fairly large), we can assume that the sample means will follow a normal distribution.
To calculate the probability, we need to find the z-score for a difference of 0.5 gallons and then use the standard normal distribution to find the area under the curve beyond that z-score. The formula for calculating the z-score is:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
Using the given values:
Population mean = 103 gallons
Standard deviation = 6 gallons
Sample size = 39 cars
Calculating the z-score:
z = (103 - 103.5) / (6 / sqrt(39))
z = -0.5 / (6 / 6.2449979984)
z = -0.5 / 1.00669444444
z ≈ -0.496528
Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of -0.496528. The probability is 0.3082.
However, since we want the probability that the sample mean would differ from the population mean by more than 0.5 gallons (in either direction), we need to find the area beyond 0.5 gallons (positive and negative), which is equivalent to finding the area beyond the absolute value of the z-score.
Therefore, the probability that the sample mean would differ from the population mean by more than 0.5 gallons is 2 * (1 - 0.3082) = 2 * 0.6918 ≈ 1.3836.