Solve the problem. Round to the nearest tenth unless indicated otherwise. The weights of certain machine components are normally distributed with a mean of 8.5 g and a standard deviation of 0.09 g. Find the two weights that separate the top 3% and the bottom 3%. These weights could serve as limits used to identify which components should be rejected. Round to the nearest hundredth of a gram.
a. 8.48 g and 8.52 g
b. 8.30 g and 8.74 g
c. 8.33 g and 8.67 g****
d. 8.46 g and 8.54 g
8.33 and 8.67
To find the weights that separate the top 3% and bottom 3% of the distribution, we need to find the z-scores associated with these percentiles, and then use them to find the corresponding weights.
Step 1: Calculate the z-scores
To find the z-scores, we'll use the z-score formula:
z = (x - μ) / σ
Where,
x = the weight we're looking for
μ = mean (8.5 g)
σ = standard deviation (0.09 g)
For the top 3%, we want to find the z-score that corresponds to the cumulative probability of 0.97 (1 - 0.03). Using a standard normal distribution table or calculator, we can find that the z-score is approximately 1.881.
For the bottom 3%, we want to find the z-score that corresponds to the cumulative probability of 0.03. Again, using a standard normal distribution table or calculator, we can find that the z-score is approximately -1.881.
Step 2: Calculate the weights
Now that we have the z-scores, we can use them to find the corresponding weights.
For the top 3%:
z = (x - μ) / σ
1.881 = (x - 8.5) / 0.09
Solving for x, we get:
x - 8.5 = 1.881 * 0.09
x - 8.5 = 0.1693
x = 8.5 + 0.1693
x ≈ 8.6693 g (rounded to the nearest hundredth)
For the bottom 3%:
z = (x - μ) / σ
-1.881 = (x - 8.5) / 0.09
Solving for x, we get:
x - 8.5 = -1.881 * 0.09
x - 8.5 = -0.1693
x = 8.5 - 0.1693
x ≈ 8.3307 g (rounded to the nearest hundredth)
Therefore, the two weights that separate the top 3% and bottom 3% are approximately 8.33 g and 8.67 g, respectively.
The correct option is c. 8.33 g and 8.67 g.
To find the weights that separate the top 3% and the bottom 3% of the distribution, we need to use the concept of z-scores.
Step 1: Convert the given weights to z-scores using the formula:
z = (x - mean) / standard deviation
For the top 3% weight, we want to find the z-score that corresponds to the area beyond the upper tail of 3%. Since the normal distribution is symmetric, the area beyond the upper tail is also 3%. Using a standard normal distribution table, we find that the z-score corresponding to an upper tail area of 0.03 is approximately 1.88.
For the bottom 3% weight, we want to find the z-score that corresponds to the area beyond the lower tail of 3%. Again, since the normal distribution is symmetric, the area beyond the lower tail is also 3%. Using the same standard normal distribution table, we find that the z-score corresponding to a lower tail area of 0.03 is approximately -1.88.
Step 2: Convert the z-scores back to weights using the formula:
x = (z * standard deviation) + mean
Calculating the top weight:
x = (1.88 * 0.09) + 8.5
x ≈ 8.67 g
Calculating the bottom weight:
x = (-1.88 * 0.09) + 8.5
x ≈ 8.33 g
Therefore, the two weights that separate the top 3% and the bottom 3% are approximately 8.33 g and 8.67 g, respectively. So the correct answer is option c.