To solve this problem, let's break down the information and approach step-by-step.

Given Information

• The diameters of bolts are normally distributed.
• Mean ($\mu$) = 0.30 inches
• Standard deviation ($\sigma$) = 0.01 inches
• We want to find the percentage of bolts with a diameter greater than 0.32 inches.

Approach

Since this follows a normal distribution, we can use the Z-score formula to convert the specific diameter value (0.32 inches) into a Z-score, which will then help us determine the probability.

Step 1: Calculate the Z-score

The Z-score formula is: $Z = \frac{X - \mu}{\sigma}$ where:

• $X$ is the value of interest (0.32 inches),
• $\mu$ is the mean (0.30 inches),
• $\sigma$ is the standard deviation (0.01 inches).

Substituting in the values: $Z = \frac{0.32 - 0.30}{0.01} = \frac{0.02}{0.01} = 2$

Step 2: Find the Probability

Using the Z-score of 2, we refer to a standard normal distribution table or use a calculator to find the probability that a bolt has a diameter greater than 0.32 inches.

For a Z-score of 2, the cumulative probability to the left is approximately 0.9772. Since we are interested in the probability of a diameter greater than 0.32 inches, we need to calculate the complement: $P(X > 0.32) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228$

Step 3: Convert to Percentage

To express this as a percentage: $0.0228 \times 100 = 2.28\%$

Conclusion

Approximately 2.28% of bolts will have a diameter greater than 0.32 inches.

Would you like further details or have any questions about this solution?

Further Questions

1. How would the answer change if the standard deviation were larger?
2. What percentage of bolts would have a diameter less than 0.28 inches?
3. How is the Z-score affected by changes in mean or standard deviation?
4. How do you interpret a Z-score in a normal distribution?
5. What would the probability be for a diameter between 0.29 and 0.31 inches?

Tip

When working with normal distributions, remember that the empirical rule can provide quick estimates: approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.