Final answer:
To find the probability that the sample mean will be greater than 403, we can use the Central Limit Theorem. Using the formula for the standard error of the sample mean, we calculate the z-score and find the corresponding probability using a standard normal distribution table or calculator. The probability is approximately 0.1151 or 11.51%.
Step-by-step explanation:
To find the probability that the sample mean will be greater than 403, we need to use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
First, we need to calculate the standard error of the sample mean, which is given by the formula: standard deviation / sqrt(sample size). In this case, the standard deviation is 25 and the sample size is 100, so the standard error is 25 / sqrt(100) = 2.5.
Next, we calculate the z-score for a sample mean of 403 using the formula: (sample mean - population mean) / standard error. The population mean is 400, so the z-score is (403 - 400) / 2.5 = 1.2.
Finally, we can use a standard normal distribution table or a calculator to find the probability corresponding to a z-score of 1.2. The probability that the sample mean will be greater than 403 is approximately 0.1151, or 11.51%.