Only D and E please ### Confidence Intervals CalculationA 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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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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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

Only D and E please

### 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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