A manufacturer knows that their items have a normally distributed length, with a mean of 8.9 inches, and standard deviation of 2.2 inches. If 2 items are chosen at random, what is the probability that their mean length is less than 7.2 inches? (Give answer to 4 decimal places.)
A manufacturer knows that their items have a normally distributed length, with a mean of 8.9 inches, and standard deviation of 2.2 inches. If 2 items are chosen at random, what is the probability that their mean length is less than 7.2 inches? (Give answer to 4 decimal places.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem Statement:**
A manufacturer knows that their items have a normally distributed length, with a mean of 8.9 inches, and standard deviation of 2.2 inches.
If 2 items are chosen at random, what is the probability that their mean length is less than 7.2 inches? (Give answer to 4 decimal places.)
**Solution:**
To find the probability, we need to calculate the Z-score for the sample mean of 7.2 inches. The Z-score is given by:
\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]
where:
- \(\bar{X}\) is the sample mean,
- \(\mu\) is the population mean,
- \(\sigma\) is the standard deviation,
- \(n\) is the sample size.
Given:
- \(\mu = 8.9\) inches,
- \(\sigma = 2.2\) inches,
- \(\bar{X} = 7.2\) inches,
- \(n = 2\).
Substituting these values into the formula gives:
\[ Z = \frac{7.2 - 8.9}{2.2 / \sqrt{2}} \]
Calculate the Z-score and then use the standard normal distribution table to find the probability corresponding to this Z-score.
Remember to round the final probability to four decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbba2c12-57b9-42d4-9dcd-e0986d5f853d%2F8587c8d3-88aa-482c-a3f4-be96c357637b%2Fg68u8ee_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A manufacturer knows that their items have a normally distributed length, with a mean of 8.9 inches, and standard deviation of 2.2 inches.
If 2 items are chosen at random, what is the probability that their mean length is less than 7.2 inches? (Give answer to 4 decimal places.)
**Solution:**
To find the probability, we need to calculate the Z-score for the sample mean of 7.2 inches. The Z-score is given by:
\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]
where:
- \(\bar{X}\) is the sample mean,
- \(\mu\) is the population mean,
- \(\sigma\) is the standard deviation,
- \(n\) is the sample size.
Given:
- \(\mu = 8.9\) inches,
- \(\sigma = 2.2\) inches,
- \(\bar{X} = 7.2\) inches,
- \(n = 2\).
Substituting these values into the formula gives:
\[ Z = \frac{7.2 - 8.9}{2.2 / \sqrt{2}} \]
Calculate the Z-score and then use the standard normal distribution table to find the probability corresponding to this Z-score.
Remember to round the final probability to four decimal places.
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