1) You run quality control for a bolt producer. This past week, the plant completed a run of 3-inch diameter bolts with mean of 3.02 inches and standard deviation of 0.04 inches. If a customer's equipment cannot use bolts greater than 3.07 inches, how many bolts of their 550,000 3-inch bolt shipment would you expect to be returned?
1) You run quality control for a bolt producer. This past week, the plant completed a run of 3-inch diameter bolts with mean of 3.02 inches and standard deviation of 0.04 inches. If a customer's equipment cannot use bolts greater than 3.07 inches, how many bolts of their 550,000 3-inch bolt shipment would you expect to be returned?
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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1. Please help me answer this including the z-table.
![### Quality Control Problem for Bolt Production
**Scenario:**
You run quality control for a bolt producer. This past week, the plant completed a run of 3-inch diameter bolts with:
- **Mean Diameter:** 3.02 inches
- **Standard Deviation:** 0.04 inches
**Problem:**
A customer’s equipment cannot use bolts greater than 3.07 inches. Given that the shipment consists of 550,000 bolts, how many of these bolts would you expect to be returned based on the specified constraint?
---
#### Explanation:
To determine the number of bolts expected to be returned, we need to calculate the probability that a randomly selected bolt exceeds the 3.07 inches threshold. This can be approached using the concept of the normal distribution, where the diameter of the bolts is assumed to follow a normal distribution with the given mean and standard deviation.
1. **Calculate the Z-score:**
The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated as:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is the value of interest (3.07 inches), \( \mu \) is the mean (3.02 inches), and \( \sigma \) is the standard deviation (0.04 inches).
2. **Determine the Probability:**
Using the Z-score, refer to the standard normal distribution table (Z-table) to find the probability that a bolt's diameter is greater than 3.07 inches.
3. **Calculate the Expected Number of Returned Bolts:**
Multiply the total number of bolts (550,000) by the probability obtained in the previous step to find the expected number of bolts to be returned.
#### Calculation Steps:
1. **Calculate the Z-score:**
\[
Z = \frac{3.07 - 3.02}{0.04} = 1.25
\]
2. **Find the Probability from Z-table:**
The Z-score of 1.25 corresponds to a probability of \( P(Z < 1.25) \approx 0.8944 \). Therefore, the probability that a bolt exceeds 3.07 inches is:
\[
P(Z > 3.07) = 1 - P(Z < 3.07) = 1 - 0.8944 =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabd5e741-0235-409f-80de-883f5f0e5da7%2F64f2b5c0-188f-4f7a-86bd-b07956bf7c58%2F5k76045_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quality Control Problem for Bolt Production
**Scenario:**
You run quality control for a bolt producer. This past week, the plant completed a run of 3-inch diameter bolts with:
- **Mean Diameter:** 3.02 inches
- **Standard Deviation:** 0.04 inches
**Problem:**
A customer’s equipment cannot use bolts greater than 3.07 inches. Given that the shipment consists of 550,000 bolts, how many of these bolts would you expect to be returned based on the specified constraint?
---
#### Explanation:
To determine the number of bolts expected to be returned, we need to calculate the probability that a randomly selected bolt exceeds the 3.07 inches threshold. This can be approached using the concept of the normal distribution, where the diameter of the bolts is assumed to follow a normal distribution with the given mean and standard deviation.
1. **Calculate the Z-score:**
The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated as:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is the value of interest (3.07 inches), \( \mu \) is the mean (3.02 inches), and \( \sigma \) is the standard deviation (0.04 inches).
2. **Determine the Probability:**
Using the Z-score, refer to the standard normal distribution table (Z-table) to find the probability that a bolt's diameter is greater than 3.07 inches.
3. **Calculate the Expected Number of Returned Bolts:**
Multiply the total number of bolts (550,000) by the probability obtained in the previous step to find the expected number of bolts to be returned.
#### Calculation Steps:
1. **Calculate the Z-score:**
\[
Z = \frac{3.07 - 3.02}{0.04} = 1.25
\]
2. **Find the Probability from Z-table:**
The Z-score of 1.25 corresponds to a probability of \( P(Z < 1.25) \approx 0.8944 \). Therefore, the probability that a bolt exceeds 3.07 inches is:
\[
P(Z > 3.07) = 1 - P(Z < 3.07) = 1 - 0.8944 =
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