Use the given information to find the number of degrees of freedom, the critical values y? and y?, and the confidence interval estimate of g. It is reasonable to assume that a simple random sample has been selected I from a population with normal distribution. Nicotine in menthol cigarettes 98% confidence; n= 20, s = 0.21 mg. Click the icon to view the table of Chi-Square critical values. ! df = 19 (Type a whole number.) Pxỉ = || (Round to three decimal places as needed.)
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
I need the critical values of x2l and x2r
![**Chi-Square Distribution and Confidence Interval Estimate for Population Standard Deviation**
**Objective:**
To find the number of degrees of freedom, the critical values \(\chi^2_L\) and \(\chi^2_R\), and the confidence interval estimate of \(\sigma\) for a simple random sample selected from a population with a normal distribution.
**Given Information:**
- Nicotine in menthol cigarettes with 98% confidence
- Sample size \(n = 20\)
- Sample standard deviation \(s = 0.21\) mg
### Steps to Follow:
1. **Degrees of Freedom:**
\[
\text{df} = n - 1
\]
Where \(n = 20\), thus:
\[
\text{df} = 20 - 1 = 19
\]
(Type a whole number in the provided field)
**df =** 19 (Type a whole number.)
2. **Critical Values:**
Use the Chi-Square distribution table to find the critical values \(\chi^2_L\) and \(\chi^2_R\) for 98% confidence and 19 degrees of freedom. Critical values are dependent on the selected confidence level and degrees of freedom.
\[
\chi^2_L =
\]
(Find this value from the Chi-Square distribution table)
\[
\chi^2_R =
\]
(Find this value from the Chi-Square distribution table)
\[
\chi^2 =
\]
(Round your answer to three decimal places as needed.)
3. **Confidence Interval Formula:**
\[
\left( \frac{(n-1)s^2}{\chi^{2}_R}, \frac{(n-1)s^2}{\chi^{2}_L} \right)
\]
where \(\chi^2_L\) and \(\chi^2_R\) are the critical values found from the Chi-Square distribution table.
#### Additional Resources:
- **Table of Chi-Square Critical Values:**
Click the icon to view the Chi-Square critical values table linked above.
### Example Calculation:
For 19 degrees of freedom:
- If using a 98% confidence level, we find \(\chi^2_R\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91a1897c-6376-41f7-9c6e-a56c1adb050a%2F0a6052bc-350c-460a-8690-a5ac39a5ac0a%2Fxywcw99.jpeg&w=3840&q=75)
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