Express the confidence interval (0.412,0.789) in the form of
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.
![### Confidence Interval Expression
**Problem Statement:**
Express the confidence interval \((0.412, 0.789)\) in the form of \(\hat{p} \pm E\).
**Solution:**
To express a confidence interval \((a, b)\) in the form of \(\hat{p} \pm E\), where \(\hat{p}\) is the sample proportion and \(E\) is the margin of error, you can follow these steps:
1. **Calculate the sample proportion (\(\hat{p}\)):**
The sample proportion \(\hat{p}\) is the midpoint of the interval.
\[
\hat{p} = \frac{a + b}{2}
\]
Substituting the given values:
\[
\hat{p} = \frac{0.412 + 0.789}{2} = 0.6005
\]
2. **Calculate the margin of error (E):**
The margin of error \(E\) is half the width of the interval.
\[
E = \frac{b - a}{2}
\]
Substituting the given values:
\[
E = \frac{0.789 - 0.412}{2} = 0.1885
\]
So, the confidence interval \((0.412, 0.789)\) can be expressed in the form:
\[
\hat{p} \pm E = 0.6005 \pm 0.1885
\]
For a visual representation, a confidence interval \( (a, b) \) can be analyzed using the center and margin of error format, which helps to interpret the range in which the true population parameter is expected to fall.
This format is particularly useful in statistics and research to summarize the precision of an estimate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b16e79-59da-42ec-8251-0c12184e0a83%2F552189ea-7f14-47e8-a5fd-76d484756b8b%2Fs8jnr36.jpeg&w=3840&q=75)
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