The lengths of lumber a machine cuts are normally distributed with a mean of 91 inches and a standard deviation of 0.4 inch. (a) What is the probability that a randomly selected board cut by the machine has a length greater than 91.15 inches? (b) A sample of 44 boards is randomly selected. What is the probability that their mean length is greater than 91.15 inches? (a) The probability is. (Round to four decimal places as needed.)
The lengths of lumber a machine cuts are normally distributed with a mean of 91 inches and a standard deviation of 0.4 inch. (a) What is the probability that a randomly selected board cut by the machine has a length greater than 91.15 inches? (b) A sample of 44 boards is randomly selected. What is the probability that their mean length is greater than 91.15 inches? (a) The probability is. (Round to four decimal places as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Understanding Probability with Normal Distribution**
**Problem Statement:**
The lengths of lumber a machine cuts are normally distributed with:
- Mean (\(\mu\)) = 91 inches
- Standard Deviation (\(\sigma\)) = 0.4 inch
**Tasks:**
(a) Determine the probability that a randomly selected board cut by the machine has a length greater than 91.15 inches.
(b) A sample of 44 boards is randomly selected. Find the probability that their mean length is greater than 91.15 inches.
**Solution Guide:**
(a) **Calculating Probability for an Individual Board:**
To find the probability that an individual board's length is greater than 91.15 inches, we use the properties of the normal distribution:
1. Calculate the Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is 91.15 inches.
2. Use a standard normal distribution table or calculator to find the probability corresponding to the calculated Z-score.
**(b) Calculating Probability for the Mean of a Sample:**
1. Use the Central Limit Theorem, which states that the distribution of the sample means will be normal or nearly normal if the sample size is large enough (n ≥ 30 is a common rule of thumb).
2. Calculate the standard error (SE) of the mean using the formula:
\[
SE = \frac{\sigma}{\sqrt{n}}
\]
where \( n = 44 \).
3. Calculate the Z-score for the sample mean:
\[
Z = \frac{\overline{X} - \mu}{SE}
\]
where \( \overline{X} \) is 91.15 inches.
4. Use a standard normal distribution table or calculator to find the probability corresponding to the calculated Z-score.
**Answer Section:**
(a) The probability is \(\_\_ \).
(Round to four decimal places as needed.)
**Additional Resources:**
- Check online calculators for normal distribution
- Review the properties of normal and standard normal distributions
- Understanding Z-scores and their application in statistical analysis
For further help and interactive tools, visit [Educational Resource](#).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb153809b-6548-4767-944e-20bfb299f2a4%2Fbed9f10b-605a-4f22-a7a7-04f32f81fc9a%2Fa6oocq6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Probability with Normal Distribution**
**Problem Statement:**
The lengths of lumber a machine cuts are normally distributed with:
- Mean (\(\mu\)) = 91 inches
- Standard Deviation (\(\sigma\)) = 0.4 inch
**Tasks:**
(a) Determine the probability that a randomly selected board cut by the machine has a length greater than 91.15 inches.
(b) A sample of 44 boards is randomly selected. Find the probability that their mean length is greater than 91.15 inches.
**Solution Guide:**
(a) **Calculating Probability for an Individual Board:**
To find the probability that an individual board's length is greater than 91.15 inches, we use the properties of the normal distribution:
1. Calculate the Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is 91.15 inches.
2. Use a standard normal distribution table or calculator to find the probability corresponding to the calculated Z-score.
**(b) Calculating Probability for the Mean of a Sample:**
1. Use the Central Limit Theorem, which states that the distribution of the sample means will be normal or nearly normal if the sample size is large enough (n ≥ 30 is a common rule of thumb).
2. Calculate the standard error (SE) of the mean using the formula:
\[
SE = \frac{\sigma}{\sqrt{n}}
\]
where \( n = 44 \).
3. Calculate the Z-score for the sample mean:
\[
Z = \frac{\overline{X} - \mu}{SE}
\]
where \( \overline{X} \) is 91.15 inches.
4. Use a standard normal distribution table or calculator to find the probability corresponding to the calculated Z-score.
**Answer Section:**
(a) The probability is \(\_\_ \).
(Round to four decimal places as needed.)
**Additional Resources:**
- Check online calculators for normal distribution
- Review the properties of normal and standard normal distributions
- Understanding Z-scores and their application in statistical analysis
For further help and interactive tools, visit [Educational Resource](#).
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