Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $156,000. Assume the standard deviation is $30,000. Suppose you take a simple random sample of 11 graduates. Round all answers to four decimal places if necessary. a. What is the distribution of X? X ~ N( b. What is the distribution of ? ~ N( c. For a single randomly selected graduate, find the probability that her salary is between $148,332 and $155,155. d. For a simple random sample of 11 graduates, find the probability that the average salary is between $148,332 and $155,155. e. For part d), is the assumption of normal necessary? O No Yes > Next Question
Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $156,000. Assume the standard deviation is $30,000. Suppose you take a simple random sample of 11 graduates. Round all answers to four decimal places if necessary. a. What is the distribution of X? X ~ N( b. What is the distribution of ? ~ N( c. For a single randomly selected graduate, find the probability that her salary is between $148,332 and $155,155. d. For a simple random sample of 11 graduates, find the probability that the average salary is between $148,332 and $155,155. e. For part d), is the assumption of normal necessary? O No Yes > Next Question
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
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![### Statistical Analysis of MBA Graduates' Salaries
Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $156,000. Assume the standard deviation is $30,000. Suppose you take a simple random sample of 11 graduates. Round all answers to four decimal places if necessary.
a. What is the distribution of \( X \)?
\[ X \sim N(\boxed{156,000}, \boxed{30,000}) \]
b. What is the distribution of \( \bar{X} \)?
\[ \bar{X} \sim N\left(\boxed{156,000}, \boxed{\frac{30,000}{\sqrt{11}}}\right) \]
c. For a single randomly selected graduate, find the probability that her salary is between $148,332 and $155,155.
\[ \boxed{\ } \]
d. For a simple random sample of 11 graduates, find the probability that the average salary is between $148,332 and $155,155.
\[ \boxed{\ } \]
e. For part d), is the assumption of normal necessary?
\[ \boxed{\text{No}} \quad \boxed{\text{Yes}} \]
### Explanation
In the context of this problem, \( X \) represents the salary of a single MBA graduate 10 years after graduation, and \( \bar{X} \) represents the average salary of a sample of 11 MBA graduates.
The normal distribution assumption helps in determining the probabilities for parts c) and d). Calculations will involve finding the area under the normal curve between specific salary values.
**Diagrams/Graphs:**
There are no diagrams or graphs in this problem statement.
**Notes:**
- \( N(\mu, \sigma) \) denotes a normal distribution with mean \( \mu \) and standard deviation \( \sigma \).
- The standard deviation of the sample mean, \( \bar{X} \), is the population standard deviation divided by the square root of the sample size.
- For part e), normality assumption is questioned as it's crucial for accurate probability calculation when the sample size is not large.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa28b18f-3e2d-4592-9d09-7b963ee49630%2Ffe18d04a-0b7e-4557-8262-1e2aab0ec5a4%2Fjnbcs7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Statistical Analysis of MBA Graduates' Salaries
Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $156,000. Assume the standard deviation is $30,000. Suppose you take a simple random sample of 11 graduates. Round all answers to four decimal places if necessary.
a. What is the distribution of \( X \)?
\[ X \sim N(\boxed{156,000}, \boxed{30,000}) \]
b. What is the distribution of \( \bar{X} \)?
\[ \bar{X} \sim N\left(\boxed{156,000}, \boxed{\frac{30,000}{\sqrt{11}}}\right) \]
c. For a single randomly selected graduate, find the probability that her salary is between $148,332 and $155,155.
\[ \boxed{\ } \]
d. For a simple random sample of 11 graduates, find the probability that the average salary is between $148,332 and $155,155.
\[ \boxed{\ } \]
e. For part d), is the assumption of normal necessary?
\[ \boxed{\text{No}} \quad \boxed{\text{Yes}} \]
### Explanation
In the context of this problem, \( X \) represents the salary of a single MBA graduate 10 years after graduation, and \( \bar{X} \) represents the average salary of a sample of 11 MBA graduates.
The normal distribution assumption helps in determining the probabilities for parts c) and d). Calculations will involve finding the area under the normal curve between specific salary values.
**Diagrams/Graphs:**
There are no diagrams or graphs in this problem statement.
**Notes:**
- \( N(\mu, \sigma) \) denotes a normal distribution with mean \( \mu \) and standard deviation \( \sigma \).
- The standard deviation of the sample mean, \( \bar{X} \), is the population standard deviation divided by the square root of the sample size.
- For part e), normality assumption is questioned as it's crucial for accurate probability calculation when the sample size is not large.
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