An automotive engineer wants to use 64 test runs to estimate the true mean tire stopping distance. The standard deviation is assumed to be 32 feet. What is the probability that the error of the estimate will be more than 4 feet?
To find the probability that the error of the estimate will be more than 4 feet, we need to calculate the probability that the sample mean is more than 4 feet away from the true mean.
Given that the standard deviation is 32 feet and the sample size is 64, we can use the Central Limit Theorem to approximate the distribution of the sample mean. According to the Central Limit Theorem, the distribution of the sample mean will be approximately normal with a mean equal to the true mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population standard deviation is 32 feet and the sample size is 64. Therefore, the standard deviation of the sample mean is 32 / sqrt(64) = 4 feet.
Now, we need to find the probability that the sample mean is more than 4 feet away from the true mean. To do this, we can calculate the z-score for a difference of 4 feet using the formula:
z = (sample mean - true mean) / standard deviation
In this case, the sample mean is unknown, but we can use the sample size to estimate the standard deviation. The standard deviation of the sample mean is 4 feet, so the standard error is also 4 feet. Therefore, the z-score can be calculated as:
z = 4 / 4 = 1
Next, we need to find the probability of z being greater than 1. We can use a standard normal distribution table or a statistical software to find this probability. Using a standard normal distribution table, we can find that the probability of z being greater than 1 is approximately 0.1587.
Therefore, the probability that the error of the estimate will be more than 4 feet is approximately 0.1587 or 15.87%.