Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 40 minutes and standard deviation 21 minutes. A researcher observed 48 students who entered the library to study. Round all answers to 4 decimal places where possible. a. What is the distribution of X X- N( b. What is the distribution of a? a- N( c. What is the distribution of x? x - N
Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 40 minutes and standard deviation 21 minutes. A researcher observed 48 students who entered the library to study. Round all answers to 4 decimal places where possible. a. What is the distribution of X X- N( b. What is the distribution of a? a- N( c. What is the distribution of x? x - N
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![The text discusses a statistical analysis involving study times of students in a library. Here’s the transcription and description:
---
### Educational Exercise: Analyzing Study Times
**Scenario:**
The study times of students in the library during one sitting are normally distributed with a mean of 40 minutes and a standard deviation of 21 minutes. A researcher observes 48 students entering the library. All answers should be rounded to four decimal places where possible.
**Questions:**
a. What is the distribution of \( X \)?
- \( X \sim N(40, 21) \)
b. What is the distribution of \( \bar{x} \)?
- \( \bar{x} \sim N(40, \frac{21}{\sqrt{48}}) \)
c. What is the distribution of \( \sum x \)?
- \( \sum x \sim N(1920, 147) \)
d. If one randomly selected student is timed, find the probability that this student’s time will be between 35 and 37 minutes.
- Answer in the box provided
e. For the 48 students, find the probability that their average time studying is between 35 and 37 minutes.
- Answer in the box provided
f. Find the probability that the randomly selected 48 students will have a total study time more than 2064 minutes.
- Answer in the box provided
g. Is the assumption of normalcy necessary for parts e) and f)?
- Yes / No (circle the correct answer)
h. The top 15% of the total study time for groups of 48 students will receive a sticker that says “Great dedication.” Determine the least time required to receive a sticker.
- Answer in the box provided
**Hints:**
- Utilize these helpful video resources to guide you through the problem:
- Finding the Sampling Distribution
- Finding a Probability Using the Central Limit Theorem
- Finding Value Given a Probability Using the Central Limit Theorem
- The Central Limit Theorem for Sums
Make sure to watch these videos for a comprehensive understanding and successful completion of the exercise. Once finished, submit your answers using the 'Submit Question' button.
---
This educational resource is designed to deepen understanding of the Central Limit Theorem and probability distributions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24b8115f-9a19-47b6-814b-3be924127cef%2Ff8299ddb-75a0-498a-897f-2eed4a37762c%2Fevfhsqr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The text discusses a statistical analysis involving study times of students in a library. Here’s the transcription and description:
---
### Educational Exercise: Analyzing Study Times
**Scenario:**
The study times of students in the library during one sitting are normally distributed with a mean of 40 minutes and a standard deviation of 21 minutes. A researcher observes 48 students entering the library. All answers should be rounded to four decimal places where possible.
**Questions:**
a. What is the distribution of \( X \)?
- \( X \sim N(40, 21) \)
b. What is the distribution of \( \bar{x} \)?
- \( \bar{x} \sim N(40, \frac{21}{\sqrt{48}}) \)
c. What is the distribution of \( \sum x \)?
- \( \sum x \sim N(1920, 147) \)
d. If one randomly selected student is timed, find the probability that this student’s time will be between 35 and 37 minutes.
- Answer in the box provided
e. For the 48 students, find the probability that their average time studying is between 35 and 37 minutes.
- Answer in the box provided
f. Find the probability that the randomly selected 48 students will have a total study time more than 2064 minutes.
- Answer in the box provided
g. Is the assumption of normalcy necessary for parts e) and f)?
- Yes / No (circle the correct answer)
h. The top 15% of the total study time for groups of 48 students will receive a sticker that says “Great dedication.” Determine the least time required to receive a sticker.
- Answer in the box provided
**Hints:**
- Utilize these helpful video resources to guide you through the problem:
- Finding the Sampling Distribution
- Finding a Probability Using the Central Limit Theorem
- Finding Value Given a Probability Using the Central Limit Theorem
- The Central Limit Theorem for Sums
Make sure to watch these videos for a comprehensive understanding and successful completion of the exercise. Once finished, submit your answers using the 'Submit Question' button.
---
This educational resource is designed to deepen understanding of the Central Limit Theorem and probability distributions.
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