Final answer:
The probability that the mean of sample 1 is greater than the mean of sample 2 is calculated using the properties of the normal distribution. After standardizing the distribution of the difference between sample means, we find a probability of approximately 0.8413, which is not among the given options.
Step-by-step explanation:
To calculate the probability that the mean of sample 1 is greater than the mean of sample 2, we must examine the sampling distribution of the difference between two means. When dealing with normal populations and large enough sample sizes, the distribution of the difference between sample means will also be normally distributed. In this case, we are considering the sampling distribution of (X1 bar - X2 bar), where X1 bar and X2 bar are the sample means of population 1 and population 2, respectively.
The mean of the sampling distribution of the difference between sample means, (μ1 - μ2), is simply the difference between the population means, so (40 - 38 = 2). The standard deviation of this sampling distribution can be found using the formula:
σd = sqrt((σ1^2/n1) + (σ2^2/n2)), where σ1 and σ2 are the standard deviations of the two populations, and n1 and n2 are the sample sizes drawn from each population.
Plugging in our values we get: σd = sqrt((6^2/25) + (8^2/25)) = sqrt((36/25) + (64/25)) = sqrt(100/25) = sqrt(4) = 2.
With these values, we can standardize the variable (X1 bar - X2 bar) and use a standard normal table (or technology) to find the probability that the difference is greater than 0. This can be represented by the following:
P(X1 bar - X2 bar > 0) = P(Z > (0 - (μ1 - μ2)) / σd)
= P(Z > (0 - 2) / 2)
= P(Z > -1).
For a Z-score of -1, the standard normal table tells us that we have a probability of approximately 0.8413 that a Z-score is greater than -1. This means that the probability that the mean of sample 1 is greater than the mean of sample 2 is approximately 0.8413, which is not given in the options provided. However, there could be a typo in the question, or a miscalculation on the student's part. Always check the calculations and the interpretation of the question.