Taxes: The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $5000. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law. Part: 0/5 Part 1 of 5 (a) What is the probability that the sample mean tax is less than $7900? Round the answer to at least four decimal places. The probability that the sample mean tax is less than $7900 is X
Taxes: The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $5000. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law. Part: 0/5 Part 1 of 5 (a) What is the probability that the sample mean tax is less than $7900? Round the answer to at least four decimal places. The probability that the sample mean tax is less than $7900 is X
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![### Educational Resource: Probability and Statistics - Federal Income Tax Example
**Topic: Calculating Probability Using Sample Mean**
---
#### Context:
The Internal Revenue Service (IRS) reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $5000. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law.
---
#### Problem Statement:
**Part 1 of 5: What is the probability that the sample mean tax is less than $7900?**
---
**Solution Steps:**
1. **Given Data:**
- Population mean (μ) = $8040
- Population standard deviation (σ) = $5000
- Sample size (n) = 1000
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{5000}{\sqrt{1000}}
\]
3. **Calculate the Z-Score:**
The Z-score for a sample mean can be calculated using the formula:
\[
Z = \frac{\bar{X} - \mu}{\text{SEM}}
\]
Where \(\bar{X}\) is the sample mean ($7900), \(\mu\) is the population mean ($8040), and \(\text{SEM}\) is the standard error of the mean.
4. **Determine the Probability:**
Use the Z-score to find the probability from the Z-table.
---
**Question:**
What is the probability that the sample mean tax is less than $7900? Round the answer to at least four decimal places.
**Blank Entry Field:**
The probability that the sample mean tax is less than $7900 is:
\[ \boxed{} \]
---
#### Notes:
- Enter the calculated probability in the provided box.
- Use accurate rounding to four decimal places.
- If you need to reference Z-tables, they are available in online statistics resources or statistical textbooks.
---
**Interactive Features:**
- `Skip Part` Button: Allows you to skip to the next part of the question.
- `Check` Button: Verifies your answer.
- `Save For Later` Button: Saves your progress to resume later.
- `Submit Assignment` Button: Submits your final answers](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29a07bd8-eb1c-498d-b0a8-9eb9d6fddf09%2F938c3ad9-e25a-4e36-bf22-cdad23c30245%2F2gvewkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Resource: Probability and Statistics - Federal Income Tax Example
**Topic: Calculating Probability Using Sample Mean**
---
#### Context:
The Internal Revenue Service (IRS) reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $5000. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law.
---
#### Problem Statement:
**Part 1 of 5: What is the probability that the sample mean tax is less than $7900?**
---
**Solution Steps:**
1. **Given Data:**
- Population mean (μ) = $8040
- Population standard deviation (σ) = $5000
- Sample size (n) = 1000
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{5000}{\sqrt{1000}}
\]
3. **Calculate the Z-Score:**
The Z-score for a sample mean can be calculated using the formula:
\[
Z = \frac{\bar{X} - \mu}{\text{SEM}}
\]
Where \(\bar{X}\) is the sample mean ($7900), \(\mu\) is the population mean ($8040), and \(\text{SEM}\) is the standard error of the mean.
4. **Determine the Probability:**
Use the Z-score to find the probability from the Z-table.
---
**Question:**
What is the probability that the sample mean tax is less than $7900? Round the answer to at least four decimal places.
**Blank Entry Field:**
The probability that the sample mean tax is less than $7900 is:
\[ \boxed{} \]
---
#### Notes:
- Enter the calculated probability in the provided box.
- Use accurate rounding to four decimal places.
- If you need to reference Z-tables, they are available in online statistics resources or statistical textbooks.
---
**Interactive Features:**
- `Skip Part` Button: Allows you to skip to the next part of the question.
- `Check` Button: Verifies your answer.
- `Save For Later` Button: Saves your progress to resume later.
- `Submit Assignment` Button: Submits your final answers
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