Suppose x has a normal distribution with s = 1.7. A random sample of size 12 has a sample mean of x = 50. ### Statistics Problem Explanation#### Problem StatementSuppose \( x \) has a normal distribution with \( s = 1.7 \). A random sample of size 12 has a sample mean of \( \bar{x} = 50 \).#### Part (a): Conditions for Calculations**Question:** What conditions are necessary for your calculations? (Select all that apply.)- \(\square\) Uniform distribution of \( x \)- \(\checked\) \( \sigma \) is unknown- \(\square\) \( \sigma \) is known- \(\checked\) Normal distribution of \( x \)**Explanation:** For the calculations to be valid, we need to assume that the original population \( x \) follows a normal distribution and that the population standard deviation \( \sigma \) is unknown.#### Part (b): 90% Confidence Interval**Question:** Find a 90% confidence interval for \( \mu \). What is the margin of error? (Round your answers to two decimal places.)- Lower limit: [Text box for input]- Upper limit: [Text box for input]- Margin of error: **0.807** (This value is given and marked with a red cross, indicating it is incorrect)**Explanation:** To find the correct 90% confidence interval and margin of error, we typically use the formula \[ \text{Margin of Error} = t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}} \]where \( t_{\alpha/2, n-1} \) is the t-value from Student’s t-distribution, \( s \) is the sample standard deviation, and \( n \) is the sample size.#### Part (c): Interpretation**Question:** Interpret your results in the context of this problem.- \(\bigcirc\) The probability that this interval contains the true average is 0.80.- \(\checked\) There is a 90% chance that the interval includes the true average.- \(\bigcirc\) The probability that this interval contains the true average is 0.20.- \(\bigcirc\) There is a 20% chance that the interval includes the true average.**Explanation:** The correct interpretation of a 90% confidence interval is that there is a 90% chance that the computed interval from the sample data includes

Given : Suppose "X" has a normal distribution with standard deviation s = 1.7 and sample mean equal…

Answered: ) What conditions are necessary for…

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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Question

Suppose x has a normal distribution with s = 1.7. A random sample of size 12 has a sample mean of x = 50.

### Statistics Problem Explanation

#### Problem Statement
Suppose \( x \) has a normal distribution with \( s = 1.7 \). A random sample of size 12 has a sample mean of \( \bar{x} = 50 \).

#### Part (a): Conditions for Calculations
**Question:** What conditions are necessary for your calculations? (Select all that apply.)

- \(\square\) Uniform distribution of \( x \)
- \(\checked\) \( \sigma \) is unknown
- \(\square\) \( \sigma \) is known
- \(\checked\) Normal distribution of \( x \)

**Explanation:** For the calculations to be valid, we need to assume that the original population \( x \) follows a normal distribution and that the population standard deviation \( \sigma \) is unknown.

#### Part (b): 90% Confidence Interval
**Question:** Find a 90% confidence interval for \( \mu \). What is the margin of error? (Round your answers to two decimal places.)

- Lower limit: [Text box for input]
- Upper limit: [Text box for input]
- Margin of error: **0.807** (This value is given and marked with a red cross, indicating it is incorrect)

**Explanation:** To find the correct 90% confidence interval and margin of error, we typically use the formula
\[ \text{Margin of Error} = t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}} \]
where \( t_{\alpha/2, n-1} \) is the t-value from Student’s t-distribution, \( s \) is the sample standard deviation, and \( n \) is the sample size.

#### Part (c): Interpretation
**Question:** Interpret your results in the context of this problem.

- \(\bigcirc\) The probability that this interval contains the true average is 0.80.
- \(\checked\) There is a 90% chance that the interval includes the true average.
- \(\bigcirc\) The probability that this interval contains the true average is 0.20.
- \(\bigcirc\) There is a 20% chance that the interval includes the true average.

**Explanation:** The correct interpretation of a 90% confidence interval is that there is a 90% chance that the computed interval from the sample data includes

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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