To determine how many bolts from the shipment of 550,000 bolts would be expected to be returned, we can approach this problem using a normal distribution model.

The key steps are as follows:

1. Define the variables:

   • Mean of bolt diameters ($\mu$) = 3.02 inches
   • Standard deviation ($\sigma$) = 0.04 inches
   • Maximum acceptable bolt size = 3.07 inches
   • Shipment size = 550,000 bolts

2. Standardize the problem: We need to find how many bolts exceed 3.07 inches in diameter. To do this, we calculate the z-score, which represents how many standard deviations a value is away from the mean. The formula for the z-score is:

   $z = \frac{x - \mu}{\sigma}$ where:

   • $x = 3.07$ (the cutoff value for bolt size),
   • $\mu = 3.02$,
   • $\sigma = 0.04$.

   Substituting the values:

   $z = \frac{3.07 - 3.02}{0.04} = \frac{0.05}{0.04} = 1.25$

3. Find the proportion of bolts exceeding 3.07 inches: We now look up the z-score of 1.25 in the standard normal distribution table or use a calculator to find the area to the left of this z-score, which gives the proportion of bolts with diameters less than or equal to 3.07 inches. For $z = 1.25$, this area is approximately 0.8944.

   Therefore, the proportion of bolts with diameters greater than 3.07 inches is:

   $1 - 0.8944 = 0.1056$

4. Calculate the expected number of bolts to be returned: Now, multiply this proportion by the total number of bolts in the shipment:

   $0.1056 \times 550,000 = 58,080$

   So, we would expect approximately 58,080 bolts to be returned because their diameter exceeds 3.07 inches.

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Would you like a breakdown of how the normal distribution or z-scores work?

Here are five related questions you might be interested in:

1. What would happen if the standard deviation were smaller, say 0.02 inches?
2. How do you calculate the probability that a bolt is smaller than a certain diameter?
3. If the customer had a more restrictive limit of 3.05 inches, how would the result change?
4. What if the shipment was 1,000,000 bolts instead of 550,000 bolts?
5. How can we reduce the number of bolts exceeding 3.07 inches?

Tip: Understanding how to use the z-score is essential when dealing with problems involving the normal distribution. It helps you assess how typical or unusual a particular data point is compared to the mean.