The average salary for a certain profession is $80,000. Assume that the standard deviation of such salaries is $27,500. Consider a random sample of 50 people in this profession and let x represent the mean salary for the sample
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
The average salary for a certain profession is
Assume that the standard deviation of such salaries is
Consider a random sample of
people in this profession and let
represent the mean salary for the sample.
![### Understanding the Sampling Distribution of Mean Salaries
Consider exploring the average salary for a certain profession, which is $80,000, with a standard deviation of $27,500. We will explore the scenario where a random sample of 50 people from this profession are considered, and \(\bar{x}\) represents the mean salary for this sample.
#### Tasks:
##### a. Finding the Population Mean (\(\mu_{\bar{x}}\))
1. **Question:**
What is \(\mu_{\bar{x}}\)?
2. **Solution:**
\[\mu_{\bar{x}} = \_\_\_ \]
##### b. Finding the Standard Error (\(\sigma_{\bar{x}}\))
1. **Question:**
What is \(\sigma_{\bar{x}}\)?
2. **Solution:**
\[\sigma_{\bar{x}} = \_\_\_ \text{ (Round to two decimal places as needed.)} \]
##### c. Describing the Shape of the Sampling Distribution
1. **Question:**
Describe the shape of the sampling distribution of \(\bar{x}\).
2. **Options:**
- ( ) A. The shape is that of a Poisson distribution.
- ( ) B. The shape is that of a normal distribution.
- ( ) C. The shape is that of a uniform distribution.
- ( ) D. The shape is that of a binomial distribution.
##### d. Finding the \(z\)-score for \(\bar{x} = 71,000\)
1. **Question:**
Find the \(z\)-score for the value \(\bar{x} = 71,000\).
2. **Solution:**
\[ z = \_\_ \text{ (Round to two decimal places as needed.)} \]
##### e. Finding the Probability \(P(\bar{x} > 71,000)\)
1. **Question:**
Find \(P(\bar{x} > 71,000)\).
2. **Solution:**
\[ P(\bar{x} > 71,000) = \_\_\_ \text{ (Round to three decimal places as needed.)} \]
##### Explanation of Diagrams or Graphs
There are no diagrams or graphs included in this transcription request. If the problem involves visual aids such as a table of normal curve areas, it is usually displayed as](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d6ce828-7421-4579-a347-7b809a44673d%2Fad52225d-bee1-400a-b4d1-cb98b882324e%2Fifjj7of_processed.png&w=3840&q=75)
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