Question 1164883: 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 $176,000. Assume the standard deviation is $32,000. Suppose you take a simple random sample of 48 graduates. Round all answers to four decimal places if necessary.
What is the distribution of
X?X~ N(____,_____)
What is the distribution of
¯x?¯x~ N(______,_______)
For a single randomly selected graduate, find the probability that her salary is between $178,172 and $184,181.
For a simple random sample of 48 graduates, find the probability that the average salary is between $178,172 and $184,181.
For part d), is the assumption of normal necessary? No or Yes
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To solve this, we use the principles of the Normal Distribution and the **Central Limit Theorem**.
### a) What is the distribution of ?
The distribution of the individual annual salaries () follows the population mean and standard deviation.
### b) What is the distribution of ?
For a sample of size , the sampling distribution of the mean () is also normal, but with a smaller standard deviation (Standard Error).
* **Mean ():**
* **Standard Error ():**
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### c) For a single randomly selected graduate, find the probability that her salary is between $178,172 and $184,181.
We need to find . We calculate the -scores for both values:
*
*
Using a standard normal distribution table or calculator:
**Probability = 0.0738**
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### d) For a simple random sample of 48 graduates, find the probability that the average salary is between $178,172 and $184,181.
We use the Standard Error () to find the -scores for the sample mean:
*
*
Using a standard normal distribution table or calculator:
**Probability = 0.2808**
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### e) For part d), is the assumption of normal necessary?
**No**.
Because the sample size () is greater than 30, the **Central Limit Theorem** ensures that the sampling distribution of the mean will be approximately normal even if the underlying population distribution is not.
Would you like me to explain how the probability would change if the sample size was smaller, say ?
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