Final answer:
To find the probability that the sample mean would differ from the population mean by more than 0.5 gallons for 39 racing cars, we can use the central limit theorem. The probability is approximately 0.3000.
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 can use the central limit theorem. According to the central limit theorem, the sampling distribution of the sample mean will be approximately normal when the sample size is large enough, regardless of the shape of the population distribution.
Since the sample size is 39, which is considered large enough, we can assume that the sampling distribution of the sample mean follows a normal distribution with the same mean as the population mean but with a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the mean of the population is 103 gallons, the standard deviation of the population is 6 gallons, and the sample size is 39. Therefore, the standard deviation of the sampling distribution is 6 / sqrt(39) = 0.9555 gallons.
To find the probability that the sample mean would differ from the population mean by more than 0.5 gallons, we need to find the probability of the sample mean being more than 0.5 gallons away from the population mean. We can calculate this using the z-score formula:
z = (sample mean - population mean) / standard deviation of sampling distribution
For a sample mean to be more than 0.5 gallons away from the population mean, the z-score would be:
z = (0.5 - 0) / 0.9555 = 0.5232
Using a standard normal distribution table (or a calculator), we can find the probability of a z-score being greater than 0.5232. The probability is approximately 0.3001 (or 0.3000 when rounded to four decimal places).