Solve the problem.
-The weights of certain machine components are normally distributed with a mean of 8.05 g and a standard deviation of 0.09 g. Find the two weights that separate the top 3% and the bottom 3%. Theses weights could serve as limits used to identify which components should be rejected.
A)7.85 g and 8.29 g
B)7.88 g and 8.22 g
C)8.01 g and 8.09 g
D)8.03 g and 8.07 g

Question

Quizplus · Accepted Answer

To find the weights that separate the top 3% and bottom 3%, we use the Z-score formula and the standard normal distribution. For the bottom 3%, we look for the Z-score corresponding to a cumulative probability of 0.03, which is approximately -1.88. For the top 3%, we look for the Z-score corresponding to a cumulative probability of 0.97 (since 1 - 0.03 = 0.97), which is approximately 1.88. Using the Z-score formula $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation, we solve for $X$ to find the weights.For the bottom 3%:$$ -1.88 = \frac{X - 8.05}{0.09} $$$$ X = -1.88 \times 0.09 + 8.05 $$$$ X \approx 7.88 \, g $$For the top 3%:$$ 1.88 = \frac{X - 8.05}{0.09} $$$$ X = 1.88 \times 0.09 + 8.05 $$$$ X \approx 8.22 \, g $$Therefore, the weights that separate the top 3% and bottom 3% are approximately 7.88 g and 8.22 g, which corresponds to choice B.

Solve the Problem. -The Weights of Certain Machine Components Are Normally Distributed with Normally