d. n= 90, x = 4.91 e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a through d? Explain. a. ( 30.511, 37.489 ) (Round to two decimal places as needed.) b. (113.985, 118.015 ) (Round to two decimal places as needed.) c. ( 12.399 , 17.601) (Round to two decihal places as needed.) d. (Round to two decimal places as needed.)
d. n= 90, x = 4.91 e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a through d? Explain. a. ( 30.511, 37.489 ) (Round to two decimal places as needed.) b. (113.985, 118.015 ) (Round to two decimal places as needed.) c. ( 12.399 , 17.601) (Round to two decihal places as needed.) d. (Round to two decimal places as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Confidence Intervals Calculation
A random sample of \( n \) measurements was selected from a population with an unknown mean \( \mu \) and standard deviation \( \sigma = 15 \). For each of the given situations (a through d), calculate a 90% confidence interval for \( \mu \).
#### Given Data:
- **a.** \( n = 50, \overline{x} = 34 \)
- **b.** \( n = 150, \overline{x} = 116 \)
- **c.** \( n = 90, \overline{x} = 15 \)
- **d.** \( n = 90, \overline{x} = 4.91 \)
#### Calculated 90% Confidence Intervals:
- **a.** \( (30.511, 37.489) \)
- *Rounded to two decimal places as needed:*
- \( (30.51, 37.49) \)
- **b.** \( (113.985, 118.015) \)
- *Rounded to two decimal places as needed:*
- \( (113.99, 118.02) \)
- **c.** \( (12.399, 17.601) \)
- *Rounded to two decimal places as needed:*
- \( (12.40, 17.60) \)
- **d.**
- *Parts of the interval are left blank; please ensure to complete the calculations as necessary.*
- \( (?, ?) \)
#### Additional Question:
- **e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a through d? Explain.**
### Explanation of Calculation Method:
To calculate a 90% confidence interval for the population mean \( \mu \) when the population standard deviation \( \sigma \) is known, the following formula is used:
\[ \overline{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \overline{x} \) is the sample mean.
- \( \sigma \) is the population standard deviation.
- \( n \) is the sample size.
- \( Z \) is the Z-score corresponding to the desired confidence level (for 90%, \( Z \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb25aa5e3-8edb-4708-bfd5-abf65dd8da34%2F4cc0d3ba-ab3c-4412-95bf-8f25baa56fe1%2Fq3di37p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Confidence Intervals Calculation
A random sample of \( n \) measurements was selected from a population with an unknown mean \( \mu \) and standard deviation \( \sigma = 15 \). For each of the given situations (a through d), calculate a 90% confidence interval for \( \mu \).
#### Given Data:
- **a.** \( n = 50, \overline{x} = 34 \)
- **b.** \( n = 150, \overline{x} = 116 \)
- **c.** \( n = 90, \overline{x} = 15 \)
- **d.** \( n = 90, \overline{x} = 4.91 \)
#### Calculated 90% Confidence Intervals:
- **a.** \( (30.511, 37.489) \)
- *Rounded to two decimal places as needed:*
- \( (30.51, 37.49) \)
- **b.** \( (113.985, 118.015) \)
- *Rounded to two decimal places as needed:*
- \( (113.99, 118.02) \)
- **c.** \( (12.399, 17.601) \)
- *Rounded to two decimal places as needed:*
- \( (12.40, 17.60) \)
- **d.**
- *Parts of the interval are left blank; please ensure to complete the calculations as necessary.*
- \( (?, ?) \)
#### Additional Question:
- **e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a through d? Explain.**
### Explanation of Calculation Method:
To calculate a 90% confidence interval for the population mean \( \mu \) when the population standard deviation \( \sigma \) is known, the following formula is used:
\[ \overline{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \overline{x} \) is the sample mean.
- \( \sigma \) is the population standard deviation.
- \( n \) is the sample size.
- \( Z \) is the Z-score corresponding to the desired confidence level (for 90%, \( Z \approx
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