Final answer:
To find the probability that the sample mean would differ from the population mean by more than 2.1 millimeters, we can use the Central Limit Theorem and the z-score. The probability is approximately 0.0058.
Step-by-step explanation:
To find the probability that the sample mean would differ from the population mean by greater than 2.1 millimeters, we can use the Central Limit Theorem which states that the distribution of sample means approaches a normal distribution as the sample size increases. The mean of the sample means is equal to the population mean, and the standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size.
In this case, the mean diameter of the steel bolts is 136 millimeters, and the standard deviation is 7 millimeters. The sample size is 35.
To find the probability, we need to calculate the z-score which represents the number of standard deviations the sample mean is away from the population mean. The formula to calculate the z-score is: z = (sample mean - population mean) / (population standard deviation / sqrt(sample size)).
Using the given values:
z = (2.1 - 0) / (7 / sqrt(35)) = 2.5322
Next, we can use a standard normal distribution table or a calculator to find the probability that the z-score is greater than 2.5322. The probability can be found by subtracting the cumulative probability from 0.5. The result is the probability that the sample mean would differ from the population mean by greater than 2.1 millimeters. Rounding to four decimal places, the probability is approximately 0.0058.