An adviser in student services would like to estimate the average monthly car payment of all IRSC students. From past research, it is known that the standard deviation of the distribution of all IRSC student car payments is $41. Determine the sample size necessary such that the margin of error of the estimate for a 99% confidence interval for the average monthly car payment of all IRSC students is at most $6.29. Round the solution up to the nearest whole number. N=
An adviser in student services would like to estimate the average monthly car payment of all IRSC students. From past research, it is known that the standard deviation of the distribution of all IRSC student car payments is $41. Determine the sample size necessary such that the margin of error of the estimate for a 99% confidence interval for the average monthly car payment of all IRSC students is at most $6.29. Round the solution up to the nearest whole number. N=
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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ISBN:9780079039897
Author:Carter
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Chapter10: Statistics
Section: Chapter Questions
Problem 25SGR
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![### Sample Size Determination for Estimating Average Monthly Car Payments of IRSC Students
An adviser in student services is tasked with estimating the average monthly car payment for all IRSC (Indian River State College) students. Based on past research, it is known that the standard deviation of the distribution of all IRSC student car payments is $41.
The objective is to determine the necessary sample size such that the margin of error for the estimate within a 99% confidence interval is at most $6.29. The solution should be rounded up to the nearest whole number.
\[ n = \_\_\_\_\_\_ \]
**Steps to Calculate Sample Size:**
1. **Identify the Standard Deviation (σ):**
\[ \sigma = 41 \]
2. **Determine the Margin of Error (E):**
\[ E = 6.29 \]
3. **Find the Z-value for 99% Confidence Level (Z):**
- For a 99% confidence level, the Z-value (Z) is approximately 2.576 (you can find this value using a Z-table).
4. **Sample Size Formula:**
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
5. **Plug in the Values:**
\[ n = \left( \frac{2.576 \cdot 41}{6.29} \right)^2 \]
6. **Calculate the Sample Size:**
\[
\begin{align*}
n & = \left( \frac{105.616}{6.29} \right)^2 \\
n & = \left( 16.8 \right)^2 \\
n & = 282.24
\end{align*}
\]
Round up 282.24 to the nearest whole number:
\[ n = 283 \]
Therefore, to estimate the average monthly car payment of all IRSC students with a margin of error of $6.29 at a 99% confidence level, a sample size of 283 students is required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F090c748e-ae6e-4835-a77f-8d1d8bf06681%2Fb81e4cdd-22dd-4814-965f-e7738cb4898c%2Fr9ee2l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Sample Size Determination for Estimating Average Monthly Car Payments of IRSC Students
An adviser in student services is tasked with estimating the average monthly car payment for all IRSC (Indian River State College) students. Based on past research, it is known that the standard deviation of the distribution of all IRSC student car payments is $41.
The objective is to determine the necessary sample size such that the margin of error for the estimate within a 99% confidence interval is at most $6.29. The solution should be rounded up to the nearest whole number.
\[ n = \_\_\_\_\_\_ \]
**Steps to Calculate Sample Size:**
1. **Identify the Standard Deviation (σ):**
\[ \sigma = 41 \]
2. **Determine the Margin of Error (E):**
\[ E = 6.29 \]
3. **Find the Z-value for 99% Confidence Level (Z):**
- For a 99% confidence level, the Z-value (Z) is approximately 2.576 (you can find this value using a Z-table).
4. **Sample Size Formula:**
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
5. **Plug in the Values:**
\[ n = \left( \frac{2.576 \cdot 41}{6.29} \right)^2 \]
6. **Calculate the Sample Size:**
\[
\begin{align*}
n & = \left( \frac{105.616}{6.29} \right)^2 \\
n & = \left( 16.8 \right)^2 \\
n & = 282.24
\end{align*}
\]
Round up 282.24 to the nearest whole number:
\[ n = 283 \]
Therefore, to estimate the average monthly car payment of all IRSC students with a margin of error of $6.29 at a 99% confidence level, a sample size of 283 students is required.
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