91Ó°ÊÓ

An experiment consists of asking your friends if they would like to raise money for a cancer association. Assuming half of your friends would agree to raise money, construct the sampling distribution of the sample proportion of affirmative answers obtained for a sample of: a. One friend. (Hint: Find the possible sample proportion values and their probabilities) b. Two friends. (Hint: The possible sample proportion values are \(0,0.50,\) and \(1.0 .\) What are their probabilities?) c. Three friends. (Hint: There are 4 possible sample proportion values.) d. Refer to parts a-c. Sketch the sampling distributions and describe how the shape is changing as the number of friends \(n\) increases.

Step by step solution

01

Understand the Task

We need to construct sampling distributions for the proportion of affirmative answers from samples of 1, 2, and 3 friends, assuming each friend has a 50% probability of agreeing to raise money. Sampling distribution refers to the probability distribution of a statistic, like a proportion, from a sample.

02

Sampling Distribution for One Friend

When sampling one friend, there are two possible outcomes: the friend agrees or disagrees. The probability of agreement (affirmative answer) is 0.5. Thus, the sampling distribution has one possible sample proportion value of 1 with probability 0.5, and 0 with probability 0.5.

03

Sampling Distribution for Two Friends

For two friends, possible outcomes are: 0 friends agree (00), 1 friend agrees (01, 10), 2 friends agree (11). The probability of each friend agreeing is 0.5. Thus: - 0 agrees: Proportion = 0, Probability = (0.5)^2 = 0.25 - 1 agrees: Proportion = 0.5, Probability = 2*(0.5)^2 = 0.5 - 2 agree: Proportion = 1, Probability = (0.5)^2 = 0.25.

04

Sampling Distribution for Three Friends

For three friends, the possible agreeing configurations include: 0 friends agree (000), 1 agrees (001, 010, 100), 2 agree (011, 101, 110), 3 agree (111). - 0 agrees: Proportion = 0, Probability = (0.5)^3 = 0.125 - 1 agrees: Proportion = 1/3, Probability = 3*(0.5)^3 = 0.375 - 2 agree: Proportion = 2/3, Probability = 3*(0.5)^3 = 0.375 - 3 agree: Proportion = 1, Probability = (0.5)^3 = 0.125.

05

Sketch and Analyze the Distributions

Graph the distributions: - For 1 friend: only values are 0 and 1, with equal height. - For 2 friends: a triangle shape with frequent 0.5 results, less frequent 0 and 1. - For 3 friends: a bell shape begins forming with more possibilities around middle values. As sample size increases from 1 to 3, distribution symmetry increases and becomes more bell-shaped, showing the central limit theorem in action.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion

When we talk about sample proportion, we are referring to the ratio of successes to the total number of trials within a sample. Imagine you are asking your friends if they would participate in fundraising. The sample proportion is simply the fraction of friends who say "yes."

For example:

• In a sample of one friend, the sample proportion can either be 0 (if the friend says "no") or 1 (if he says "yes").
• In a sample of two friends, the sample proportion values can be 0 (if both say "no"), 0.5 (if one says "yes" and one says "no"), and 1 (if both say "yes").
• In a sample of three friends, you will have a wider range of proportions, such as 0, 1/3, 2/3, and 1.

Understanding these proportions helps in estimating how large groups might behave based on a smaller sample.

Probability Distribution

Probability distribution describes how probabilities are assigned to values of a random variable. In this context, it's about determining the likelihood of each sample proportion.

Take two friends as an example. The probability distribution would look like this:

• Probability of 0 (no one agrees) is: \((0.5)^2 = 0.25\)
• Probability of 0.5 (one agrees) is: \(2 imes (0.5)^2 = 0.5\)
• Probability of 1 (both agree) is: \((0.5)^2 = 0.25\)

This probability distribution provides a full picture of what we can expect when randomly sampling from our pool of friends. Each probability tells us how likely each outcome is in practical terms.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It says that, under certain conditions, the sampling distribution of the sample mean (or proportion) becomes approximately normal as the sample size increases, regardless of the shape of the population distribution.

As seen in the original exercise, when you increase the number of friends from 1 to 3, the shape of the sampling distribution starts appearing more bell-shaped (symmetrical). This transformation mirrors the CLT, where larger samples tend to result in a distribution resembling the normal distribution, showing greater uniformity and predictability.

Statistical Experiment

A statistical experiment involves a process that leads to a set of possible outcomes. In this scenario, asking friends if they would help raise money is the experiment.

Each friend independently responds "yes" or "no," which makes this a Bernoulli trial. Key characteristics of such an experiment include:

• Each trial can be categorized as a success or failure.
• The probability of success (a friend saying "yes") remains constant at 0.5.
• Trials are independent of one another, meaning one friend’s answer doesn’t influence another’s.

Understanding statistical experiments is critical for setting up models accurately and utilizing probability distributions effectively.