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Chapter 5: Problem 32

Define the term "sampling distribution" of the sample proportion, and describe how the shape, center, and spread of the sampling distribution change as the sample size increases when \(p=0.1\)

Short Answer

Expert verified
The sampling distribution is approximately normal, centered at 0.1, and becomes narrower as the sample size increases.

Step by step solution

01

Understanding Sampling Distribution

The sampling distribution of the sample proportion is the probability distribution of all possible sample proportions we could get by taking samples of a fixed size from a population. When we repeatedly take samples from a population and calculate the sample proportion for each sample, these proportions form their own distribution, which is called the sampling distribution of the sample proportion.
02

Determine Shape of the Distribution

When the sample size is large, the shape of the sampling distribution of the sample proportion is approximately normal. According to the Central Limit Theorem, for sufficiently large sample size, the shape approaches a normal distribution, even when the population proportion is not normal.
03

Calculate the Center of the Distribution

The center of the sampling distribution of the sample proportion is the mean of the distribution. It is always equal to the true population proportion, denoted as \( p \). For this exercise, since \( p = 0.1 \), the mean of the sampling distribution is also 0.1.
04

Determine the Spread of the Distribution

The spread of the sampling distribution is measured by its standard deviation, often called the standard error. This is calculated using the formula \( \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the sample size. As the sample size \( n \) increases, the standard error decreases, making the distribution tighter around the mean.
05

Effect of Increasing Sample Size

As the sample size increases, the sampling distribution becomes more narrowly concentrated around the true population proportion. This means the distribution becomes taller and narrower due to the smaller standard error, while the center remains constant at the population proportion \( p = 0.1 \).

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Key Concepts

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

Sample Proportion
The term "sample proportion" refers to the fraction of observations in a sample that possess a certain characteristic of interest. It is calculated by dividing the number of favorable outcomes by the total number of samples. For example, if you survey 100 people and find that 10 of them prefer apple juice over orange juice, then your sample proportion for apple juice preference is 0.1 or 10%.
Understanding the sample proportion is essential because it helps us make inferences about the larger population. By analyzing sample data, we can estimate what proportion of the entire population shares the characteristic seen in the sample.
  • Sample Proportion = (Number of Favorable Outcomes) / (Total Sample Size)
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the sampling distribution of the sample mean (or sample proportion) approaches a normal distribution as the sample size becomes larger, regardless of the original distribution of the population.
This theorem is crucial because it allows statisticians to make inferences about the population parameters using the normal distribution. Particularly for sample proportions, the CLT ensures that even if our population proportion is not normally distributed, the distribution of sample proportions will be approximately normal when the sample size is large enough.
  • This normal approximation is better as the sample size (n) increases.
  • Why is this helpful? Because normal distributions are easier to work with and have predictable properties.
Standard Error
The standard error is a measure of the variability or spread of a sampling distribution. Specifically, it indicates how much the sample proportion is likely to vary from the true population proportion from sample to sample.
For sample proportions, the standard error can be calculated using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \]
In this formula, \( p \) represents the population proportion, and \( n \) is the sample size. The standard error decreases as the sample size increases, reflecting that larger samples yield more precise estimates of the population proportion.
  • Smaller standard error means less spread in our sampling distribution.
  • Better precision and reliability in estimating the population parameter.
Population Proportion
The population proportion is a parameter that describes what fraction of the entire population possesses a particular trait or characteristic. Unlike the sample proportion, which is an estimate based on a subset, the population proportion is a true value for the entire group.
In practical research, we often aim to approximate the population proportion by using sample data because examining entire populations is usually impractical.
  • Denoted by \( p \) in statistics.
  • Aiming to accurately estimate \( p \) guides the design of studies and experiments.

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Most popular questions from this chapter

400 students were randomly sampled from a large university, and 289 said they did not get enough sleep. Conduct a hypothesis test to check whether this represents a statistically significant difference from \(50 \%\), and use a significance level of 0.01 .

In 2013, the Pew Research Foundation reported that " \(45 \%\) of U.S. adults report that they live with one or more chronic conditions". \(^{12}\) However, this value was based on a sample, so it may not be a perfect estimate for the population parameter of interest on its own. The study reported a standard error of about \(1.2 \%\), and a normal model may reasonably be used in this setting. Create a \(95 \%\) confidence interval for the proportion of U.S. adults who live with one or more chronic conditions. Also interpret the confidence interval in the context of the study.

The General Social Survey asked the question: "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a \(95 \%\) confidence interval of 3.40 to 4.24 days in 2010 (a) Interpret this interval in context of the data. (b) What does "95\% confident" mean? Explain in the context of the application. (c) Suppose the researchers think a \(99 \%\) confidence level would be more appropriate for this interval. Will this new interval be smaller or wider than the \(95 \%\) confidence interval? (d) If a new survey were to be done with 500 Americans, do you think the standard error of the estimate be larger, smaller, or about the same.

Suppose you conduct a hypothesis test based on a sample where the sample size is \(n=50,\) and arrive at a p-value of 0.08 . You then refer back to your notes and discover that you made a careless mistake, the sample size should have been \(n=500\). Will your p-value increase, decrease, or stay the same? Explain.

Teens were surveyed about cyberbullying, and \(54 \%\) to \(64 \%\) reported experiencing cyberbullying (95\% confidence interval). \(^{20}\) Answer the following questions based on this interval. (a) A newspaper claims that a majority of teens have experienced cyberbullying. Is this claim supported by the confidence interval? Explain your reasoning. (b) A researcher conjectured that \(70 \%\) of teens have experienced cyberbullying. Is this claim supported by the confidence interval? Explain your reasoning. (c) Without actually calculating the interval, determine if the claim of the researcher from part (b) would be supported based on a \(90 \%\) confidence interval?
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