You take a sample of 30 American adults and find the average amount of sleep that they get per night is 7.7 hours with a standard deviation of 1.12 hours. Compute a 97% confidence interval for the mean hours of sleep that American adults get per night. Please write your answers to at least four places. Assume that all conditions necessary for inference are satisfied. a. Critical value: t* = b. Standard error: c. Margin of error: d. Confidence interval:
You take a sample of 30 American adults and find the average amount of sleep that they get per night is 7.7 hours with a standard deviation of 1.12 hours. Compute a 97% confidence interval for the mean hours of sleep that American adults get per night. Please write your answers to at least four places. Assume that all conditions necessary for inference are satisfied. a. Critical value: t* = b. Standard error: c. Margin of error: d. Confidence interval:
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.3: Measures Of Spread
Problem 26PFA
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![**Title: Calculating a 97% Confidence Interval for Mean Sleep Duration of American Adults**
---
**Problem Statement:**
You take a sample of 30 American adults and find the average amount of sleep that they get per night is 7.7 hours with a standard deviation of 1.12 hours. Compute a 97% confidence interval for the mean hours of sleep that American adults get per night.
Please write your answers to at least four decimal places. Assume that all conditions necessary for inference are satisfied.
---
**Solution Steps:**
**a. Critical value: \( t* \) =**
\[ \text{[Insert Critical Value]} \]
The critical value \( t* \) is determined based on the confidence level (97%) and the degrees of freedom (n-1=29).
**b. Standard error:**
\[ \text{Standard error} = \frac{s}{\sqrt{n}} \]
Where:
- \( s \) is the standard deviation (1.12 hours)
- \( n \) is the sample size (30)
\[
\text{Standard error} = \frac{1.12}{\sqrt{30}} = \text{[Insert Standard Error]}
\]
**c. Margin of error:**
\[ \text{Margin of error} = t* \times \text{Standard error} \]
\[
\text{Margin of error} = \text{[Insert Critical Value]} \times \text{[Insert Standard Error]} = \text{[Insert Margin of Error]}
\]
**d. Confidence interval:**
The confidence interval for the mean is given by:
\[ \text{Point Estimate} \pm \text{Margin of error} \]
Where:
- The point estimate is the sample mean (7.7 hours)
\[
\text{Confidence interval:} \quad \left( \text{Point Estimate - Margin of error} \right) < \mu < \left( \text{Point Estimate + Margin of error} \right)
\]
\[
\left( 7.7 - \text{[Insert Margin of Error]} \right) < \mu < \left( 7.7 + \text{[Insert Margin of Error]} \right)
\]
Ensure each calculation step includes the appropriate number of decimal places as specified.
-----
**Explanation of Diagrams:**
There are no graphs or diagrams included with this question](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8e28faf-547f-4c48-bb6d-e40546588d1c%2Faac86084-4a77-4aad-98da-315613cf9818%2F6pu8gf9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating a 97% Confidence Interval for Mean Sleep Duration of American Adults**
---
**Problem Statement:**
You take a sample of 30 American adults and find the average amount of sleep that they get per night is 7.7 hours with a standard deviation of 1.12 hours. Compute a 97% confidence interval for the mean hours of sleep that American adults get per night.
Please write your answers to at least four decimal places. Assume that all conditions necessary for inference are satisfied.
---
**Solution Steps:**
**a. Critical value: \( t* \) =**
\[ \text{[Insert Critical Value]} \]
The critical value \( t* \) is determined based on the confidence level (97%) and the degrees of freedom (n-1=29).
**b. Standard error:**
\[ \text{Standard error} = \frac{s}{\sqrt{n}} \]
Where:
- \( s \) is the standard deviation (1.12 hours)
- \( n \) is the sample size (30)
\[
\text{Standard error} = \frac{1.12}{\sqrt{30}} = \text{[Insert Standard Error]}
\]
**c. Margin of error:**
\[ \text{Margin of error} = t* \times \text{Standard error} \]
\[
\text{Margin of error} = \text{[Insert Critical Value]} \times \text{[Insert Standard Error]} = \text{[Insert Margin of Error]}
\]
**d. Confidence interval:**
The confidence interval for the mean is given by:
\[ \text{Point Estimate} \pm \text{Margin of error} \]
Where:
- The point estimate is the sample mean (7.7 hours)
\[
\text{Confidence interval:} \quad \left( \text{Point Estimate - Margin of error} \right) < \mu < \left( \text{Point Estimate + Margin of error} \right)
\]
\[
\left( 7.7 - \text{[Insert Margin of Error]} \right) < \mu < \left( 7.7 + \text{[Insert Margin of Error]} \right)
\]
Ensure each calculation step includes the appropriate number of decimal places as specified.
-----
**Explanation of Diagrams:**
There are no graphs or diagrams included with this question
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