To solve this problem, let's use the properties of the normal distribution and the concept of the standard error of the mean. Let's go through the steps to calculate the probability that the means of two samples of 1000 components each will differ by more than 0.002 g.
The mean weight of the components is given as 0.50 g, and the standard deviation is given as 0.02 g.
Step 1: Calculate the standard error of the mean (SEM) for one sample.
The standard error of the mean is given by the formula:
\[ SEM = \frac{\sigma}{\sqrt{n}} \]
Where:
- \(\sigma\) is the standard deviation of the population (0.02 g in this case)
- \(n\) is the sample size (1000 in this case)
\[ SEM = \frac{0.02}{\sqrt{1000}} \]
Calculating this:
\[ SEM = \frac{0.02}{\sqrt{1000}} = \frac{0.02}{31.6228} = 0.000632455 \]
Step 2: Find the standard error of the difference between two independent sample means.
When comparing the means of two independent samples, the standard error of the difference between the means is the square root of the sum of the squares of the two sample SEMs. However, since both samples are of the same size and come from the same population, the standard error of the difference between the sample means is:
\[ SE_{difference} = \sqrt{SEM^2 + SEM^2} = \sqrt{2} \cdot SEM \]
\[ SE_{difference} = \sqrt{2} \cdot 0.000632455 \]
\[ SE_{difference} = 0.000894427 \]
Step 3: Calculate the z-score for the difference of 0.002 g.
The z-score tells us how many standard errors the difference of 0.002 g is away from 0. The z-score is calculated by:
\[ Z = \frac{Difference}{SE_{difference}} \]
\[ Z = \frac{0.002}{0.000894427} \]
Calculating this:
\[ Z \approx 2.236 \]
Step 4: Find the probability that the two means differ by more than 0.002 g.
Since we are looking for the probability that the sample means differ by more than 0.002 g in either direction, we use a two-tailed test. We look up the z-score in the standard normal distribution table to find the probability corresponding to Z = 2.236.
The z-score of approximately 2.236 corresponds to a tail probability of roughly 0.0125 on each side of the normal distribution curve (this is approximate; the exact value can be found using statistical software or a standard normal table). Since we want both tails, we double this probability for a two-tailed test.
Probability for one tail: 0.0125
Probability for two tails (both directions): 2 * 0.0125 = 0.025
Therefore, the probability that the means of the two samples will differ by more than 0.002 g is approximately 0.025 or 2.5%.
