Please help me asnwer both questions! ### Educational Exercise: Confidence Interval Calculation**Problem Description:**Suppose $ x $ has a normal distribution with $ \sigma = 1 $. A random sample of size 18 has a sample mean of $\bar{x} = 50$.#### (a) Conditions for CalculationsIdentify the necessary conditions for the calculations:- [ ] $ n $ is large- [ ] Uniform distribution of $ x $- [x] $ \sigma $ is unknown- [ ] $ \sigma $ is known*Note: The correct conditions include a normal distribution of $ x $ and if $ \sigma $ is known or unknown.*#### (b) Finding a 90% Confidence Interval for $\mu$Calculate and input the following values, rounding to two decimal places:- **Lower limit:** [ ]- **Upper limit:** [ ]- **Margin of error:** [ ]*Note: Ensure calculations are precise as any errors would be indicated by a red cross.*This exercise helps students understand the process of determining necessary conditions and calculating confidence intervals in statistics. Sure! Here's the transcription and explanation:---**Title: Calculating Confidence Intervals for a Normal Distribution****Problem Statement:**Suppose $ x $ has a normal distribution with $ \sigma = 11 $. A random sample of size 35 has a sample mean of $ \bar{x} = 70 $.**(a) What conditions are necessary for your calculations? (Select all that apply.)** - [ ] uniform distribution of $ x $ - [ ] $ \sigma $ is unknown - [ ] $ n $ is large - [x] normal distribution of $ x $ - [x] $ \sigma $ is known **(b) Find a 95% confidence interval for $\mu$. What is the margin of error? (Round your answers to two decimal places.)** *Note: Fill in the blanks based on your calculations.*- Lower limit: [______]- Upper limit: [______]- Margin of error: [______]---**Explanation:**In part (a), the necessary conditions for these calculations include having a normal distribution of $ x $ and knowing the population standard deviation $ \sigma $. These conditions allow the use of the Z-distribution to calculate the confidence interval.In part (b), you are asked to find the 95% confidence interval for the population mean $\mu$. The formula for the confidence interval is:\[\bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)\]With a 95% confidence level, the Z-value is typically 1.96. Calculate the margin of error and determine the lower and upper limits accordingly.

Name: Please help me asnwer both questions! ### Educational Exercise: Confid
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Description: Please help me asnwer both questions! ### Educational Exercise: Confidence Interval Calculation**Problem Description:**Suppose $ x $ has a normal distribution with $ \sigma = 1 $. A random sample of size 18 has a sample mean of $\bar{x} = 50$.#

X normally distributed standard deviation =1 sample size (n) = 18 Mean (x¯) =50 We know that For…

Answered: Suppose x has a normal distribution…

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Suppose x has a normal distribution with o = 1. A random sample of size 18 has a sample mean of x = 50.

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Please help me asnwer both questions!

### Educational Exercise: Confidence Interval Calculation

**Problem Description:**

Suppose $ x $ has a normal distribution with $ \sigma = 1 $. A random sample of size 18 has a sample mean of $\bar{x} = 50$.

#### (a) Conditions for Calculations

Identify the necessary conditions for the calculations:

- [ ] $ n $ is large
- [ ] Uniform distribution of $ x $
- [x] $ \sigma $ is unknown
- [ ] $ \sigma $ is known

*Note: The correct conditions include a normal distribution of $ x $ and if $ \sigma $ is known or unknown.*

#### (b) Finding a 90% Confidence Interval for $\mu$

Calculate and input the following values, rounding to two decimal places:

- **Lower limit:** [ ]
- **Upper limit:** [ ]
- **Margin of error:** [ ]

*Note: Ensure calculations are precise as any errors would be indicated by a red cross.*

This exercise helps students understand the process of determining necessary conditions and calculating confidence intervals in statistics.

Sure! Here's the transcription and explanation:

---

**Title: Calculating Confidence Intervals for a Normal Distribution**

**Problem Statement:**
Suppose $ x $ has a normal distribution with $ \sigma = 11 $. A random sample of size 35 has a sample mean of $ \bar{x} = 70 $.

**(a) What conditions are necessary for your calculations? (Select all that apply.)**
- [ ] uniform distribution of $ x $
- [ ] $ \sigma $ is unknown
- [ ] $ n $ is large
- [x] normal distribution of $ x $
- [x] $ \sigma $ is known

**(b) Find a 95% confidence interval for $\mu$. What is the margin of error? (Round your answers to two decimal places.)**
*Note: Fill in the blanks based on your calculations.*
- Lower limit: [______]
- Upper limit: [______]
- Margin of error: [______]

---

**Explanation:**

In part (a), the necessary conditions for these calculations include having a normal distribution of $ x $ and knowing the population standard deviation $ \sigma $. These conditions allow the use of the Z-distribution to calculate the confidence interval.

In part (b), you are asked to find the 95% confidence interval for the population mean $\mu$. The formula for the confidence interval is:
\[
\bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
With a 95% confidence level, the Z-value is typically 1.96. Calculate the margin of error and determine the lower and upper limits accordingly.

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