30. thought provoking consider a large population of which p percent (in decimal form) have a certain…

30. 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.

30. 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.

Answer

Explanation:

Step1: Analyze the function

Let (y = 1500p(1 - p)=1500(p - p^{2})), where (y) represents the right - hand side of the inequality (n>1500p(1 - p)).

Step2: Find the vertex of the quadratic function

The function (y = 1500(p - p^{2})=- 1500p^{2}+1500p) is a quadratic function in the form (y = ax^{2}+bx + c) with (a=-1500), (b = 1500), and (c = 0). The (x) - coordinate of the vertex of a quadratic function (y=ax^{2}+bx + c) is given by (p=-\frac{b}{2a}). Substituting (a=-1500) and (b = 1500) into the formula, we get (p=-\frac{1500}{2\times(-1500)} = 0.5).

Step3: Analyze the behavior of the function

Since (a=-1500<0), the graph of the quadratic function (y = 1500(p - p^{2})) is a parabola opening 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) to (0.5), the value of (y) increases, and as (p) moves from (0.5) to (1), the value of (y) decreases.

Answer:

The sample size (n) needed is related to the function (n>1500p(1 - p)). The size of the sample needed first increases and then decreases as the proportion (p) of the population with the characteristic changes from (0) to (1). The maximum value of (1500p(1 - p)) occurs when (p = 0.5), meaning that when half of the population has the characteristic, the largest sample size is required to ensure representativeness. When (p) is close to (0) or (1) (i.e., very few or almost all of the population has the characteristic), a smaller sample size is sufficient.