Most married couples have two or three personality preferences in common. A random sample of 384 married couples found that 130 had three preferences in common. Another random sample of 574 couples showed that 200 had two personality preferences in common. Let p, be the population proportion of all married couples who have three personality preferences in common. Let p, be the population proportion of all married couples who have two personality preferences in common. In USE SALT (a) Find a 95% confidence interval for p, - Pɔ. (Use 3 decimal places.) lower limit upper limit (b) Examine the confidence interval in part (a) and explain what it means in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you about the proportion of married couples with three personality preferences in common compared with the proportion of couples with two preferences in common (at the 95% confidence level)? Because the interval contains only positive numbers, we can say that a higher proportion of married couples have three personality preferences in common. Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have three personality preferences in common. We can not make any conclusions using this confidence interval. Because the interval contains only negative numbers, we can say that a higher proportion of married couples have two personality preferences in common.

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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.