Question

1. thought provoking consider a large population of which p percent (in decimal form) have a certain characteristic. to be reasonably sure that you are choosing a sample that is representative of a population, you should choose a random sample of n people where n > 1500p(1 - p). how does the percent of a population that has the characteristic affect the size of the sample needed? explain your reasoning.

Explanation:

Step1: Analyze the function

Let \(y = 1500p(1 - p)=1500(p - p^{2})\). This is a quadratic - function. The general form of a quadratic function is \(y = ax^{2}+bx + c\), where in our case \(a=-1500\), \(b = 1500\), and \(c = 0\).

Step2: Find the vertex of the quadratic function

The \(x\) - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). Substituting \(a=-1500\) and \(b = 1500\) into the formula, we get \(p=-\frac{1500}{2\times(-1500)}=\frac{1}{2}=0.5\).

Step3: Analyze the behavior of the function

Since \(a=-1500\lt0\), the quadratic function \(y = 1500p(1 - p)\) is a parabola that opens downwards. When \(p = 0\) or \(p = 1\), \(y=1500\times0\times(1 - 0)=0\) and \(y=1500\times1\times(1 - 1)=0\). As \(p\) moves from \(0\) towards \(0.5\), the value of \(y = 1500p(1 - p)\) increases, and as \(p\) moves from \(0.5\) towards \(1\), the value of \(y = 1500p(1 - p)\) decreases.

The sample - size \(n\) must satisfy \(n>1500p(1 - p)\). So, the sample size needed is maximized when \(p = 0.5\). When \(p\) is close to \(0\) or \(1\) (i.e., when very few or almost all of the population have the characteristic), the sample size needed is relatively small. When \(p\) is around \(0.5\) (i.e., when about half of the population have the characteristic), the sample size needed is relatively large.

Answer:

The sample size needed is maximized when \(p = 0.5\). When \(p\) is close to \(0\) or \(1\), the sample size needed is relatively small. When \(p\) is around \(0.5\), the sample size needed is relatively large. This is because the function \(n = 1500p(1 - p)\) is a quadratic function that opens downwards with a maximum at \(p = 0.5\).