5.4 Activity 13: Sampling Distribution of Proportions **Objective:** The purpose of this activity is to obtain a better understanding of the sampling distribution for the sample proportion. **Topics covered:** 1. Simple random sampling 2. Standard errors 3. Sampling distributions In this section we will construct empirical sampling distributions for the probability of heads in a coin toss. To begin, each student should take twenty pennies from the container. 1. First, let's think about what we expect to see in this experiment. (a) If we were to toss a handful of \( n \) pennies, what proportion would you expect to land on heads? That is, what is the value of \( p = \) theoretical probability of heads? \[ p = \_\_\_\_\_\_ \] (b) Are you guaranteed to observe the proportion of heads is exactly \( p \) when you toss the handful of coins? If we toss the same handful of \( n \) pennies several times, will we observe the same proportion of heads each toss? Explain. (c) The standard error (SE) describes how the observed proportion of heads varies from trial to trial. What is the general expression for \( SE_{\hat{p}} \)? \[ SE_{\hat{p}} = \_\_\_\_\_\_ \] (d) Consider the general expression for \( SE_{\hat{p}} \) in (c). What does this tell us about the role of sample size, \( n \)? Will our different tosses yield more or less consistent results as we increase the number of pennies in our sample? Explain. (e) According to the Central Limit Theorem, what should we expect for the shape of the sample proportions from many different tosses?
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
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D,E
![5.4 Activity 13: Sampling Distribution of Proportions
**Objective:** The purpose of this activity is to obtain a better understanding of the sampling distribution for the sample proportion.
**Topics covered:**
1. Simple random sampling
2. Standard errors
3. Sampling distributions
In this section we will construct empirical sampling distributions for the probability of heads in a coin toss. To begin, each student should take twenty pennies from the container.
1. First, let's think about what we expect to see in this experiment.
(a) If we were to toss a handful of \( n \) pennies, what proportion would you expect to land on heads? That is, what is the value of \( p = \) theoretical probability of heads?
\[
p = \_\_\_\_\_\_
\]
(b) Are you guaranteed to observe the proportion of heads is exactly \( p \) when you toss the handful of coins? If we toss the same handful of \( n \) pennies several times, will we observe the same proportion of heads each toss? Explain.
(c) The standard error (SE) describes how the observed proportion of heads varies from trial to trial. What is the general expression for \( SE_{\hat{p}} \)?
\[
SE_{\hat{p}} = \_\_\_\_\_\_
\]
(d) Consider the general expression for \( SE_{\hat{p}} \) in (c). What does this tell us about the role of sample size, \( n \)? Will our different tosses yield more or less consistent results as we increase the number of pennies in our sample? Explain.
(e) According to the Central Limit Theorem, what should we expect for the shape of the sample proportions from many different tosses?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d86676b-0eb6-42ee-b104-da46ee0f3304%2F03f94db4-730c-4739-bd3d-03bfd915ef92%2Foykme5.jpeg&w=3840&q=75)
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