A manufacturer wishes to ensure that 98% of the bolts that are produced from a manufacturing process have a diameter that lies within ±0.05 mm of the mean. For this to be so, then the standard deviation of the process must be equal to:

To solve this problem, we need to find the standard deviation (\(\sigma\)) of the manufacturing process such that 98% of the produced bolts have a diameter within ±0.05 mm of the mean. This means that the interval \(\mu \pm 0.05\) should contain 98% of the data.

### Step 1: Identify the Z-scores
The Z-scores that correspond to the middle 98% of the normal distribution can be found by considering that 1% of the distribution lies in each tail (since 2% is outside the interval, and the distribution is symmetric).

Using a Z-table or calculator, the Z-score that corresponds to the 1% in the lower tail (or 99% in the upper tail) is approximately:

\[
Z \approx 2.33
\]

### Step 2: Set up the equation
The Z-score is related to the standard deviation (\(\sigma\)) by the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Here, \(X = \mu \pm 0.05\), and we are looking for the standard deviation \(\sigma\) such that:

\[
2.33 = \frac{0.05}{\sigma}
\]

### Step 3: Solve for \(\sigma\)
Rearranging the equation to solve for \(\sigma\):

\[
\sigma = \frac{0.05}{2.33} \approx 0.0215 \, \text{mm}
\]

So, the standard deviation of the process must be approximately **0.0215 mm** to ensure that 98% of the bolts have a diameter within ±0.05 mm of the mean.

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Do you need any further explanation or have other questions? Here are five related questions you might find interesting:

1. What would the standard deviation need to be if the manufacturer wanted 99% of the bolts within ±0.05 mm?
2. If the mean diameter of the bolts is 10 mm, what range of diameters should the manufacturer expect for 95% of the bolts?
3. How would increasing the tolerance from ±0.05 mm to ±0.1 mm affect the required standard deviation?
4. What percentage of bolts would fall within ±0.03 mm of the mean if the standard deviation is 0.0215 mm?
5. How does the value of the Z-score change with different confidence levels (e.g., 95%, 99%)?

**Tip:** When dealing with process control in manufacturing, understanding the relationship between the Z-score, standard deviation, and tolerance limits is crucial for ensuring product quality.

Learn how to calculate the standard deviation required for a manufacturing process to ensure that 98% of bolts have diameters within ±0.05 mm of the mean. This problem involves statistical concepts such as the Z-score and normal distribution. Suitable for advanced high school or college-level students.

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