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Chapter 6: Problem 4

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion: (a) Find the mean and standard error of the distribution of sample proportions. (b) If the sample size is large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 30 from a population with proportion 0.27

Short Answer

Expert verified
The mean of the sample proportions is 0.27 and standard error is the square root of \((0.27 \times (1-0.27) / 30)\). Central Limit Theorem applies, so the distribution curve is approximately normal, centered at 0.27.

Step by step solution

01

Calculate the mean

The mean, denoted by \(\mu_p\), of the distribution of sample proportions is equal to the proportion of the population. Thus, \(\mu_p = 0.27\)
02

Calculate standard error

The standard error, denoted by \(SE_p\), of the distribution of sample proportions is calculated as \(\sqrt{p(1-p)/n}\), where p is the proportion of the population and n is the sample size. Thus, \(SE_p = \sqrt{0.27(1-0.27)/30}\)
03

Create distribution curve

Since the sample size is larger than 30, Central Limit Theorem can be applied which means that the shape of the sampling distribution will be approximately normally distributed. The curve will be centered at the mean 0.27, and it will spread out based on the computed standard error.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics. It tells us that if we take a large enough sample size from any population, the distribution of the sample means will be approximately normal. This holds true regardless of the original population's distribution, making it a powerful tool. This theorem allows us to make inferences about population parameters using sample statistics.

### Key Takeaways
  • Sample size matters: The CLT kicks in when you have a sufficiently large sample size. Usually, a sample size of 30 or more is considered large enough.
  • Approximate normality: Even if the population distribution is not normal, the distribution of sample means (or sample proportions) will tend to be normal if the sample size is large.
  • Application: It simplifies complex problems, allowing for easier predictions and interventions in real-world scenarios.
Understanding the CLT is crucial because it underpins many statistical methods, making analysis of data more manageable.
Sample Proportions
Sample proportions refer to the proportion of a certain characteristic within a sample taken from a population. For example, if you have a sample of 100 people and 27 have a certain trait, the sample proportion is 0.27. This is immensely useful for estimating the population parameter, often denoted as "p."

### Points to Remember
  • Representation: Sample proportions are an estimate of the population proportion, which aids in making statistical inferences.
  • Sample size's role: A larger sample size will generally produce a more accurate estimate of the population proportion.
  • Foundation for sampling distribution: This forms the basis for understanding how sample proportions distribute around the true population proportion.
Sample proportions are foundational to inferential statistics, driving the conclusions we draw from data analysis.
Standard Error
The standard error is a measure that reflects the variability of a statistic. When dealing with sample proportions, the standard error helps us understand how the sample proportion (the statistic) varies from sample to sample. It is calculated using the formula: \[ SE_p = \sqrt{\frac{p(1-p)}{n}} \] where \(p\) is the population proportion and \(n\) is the sample size.

### Importance of Standard Error
  • Predict variability: It gives insight into the expected fluctuation of sample proportions around the true population proportion.
  • Dependency on sample size: As the sample size increases, the standard error decreases, indicating more precise estimates.
  • Connection to normal distribution: It helps us understand the spread of the sampling distribution under the Central Limit Theorem.
In sum, understanding the concept of standard error is vital for gauging the accuracy and reliability of our sample statistics.

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Most popular questions from this chapter

Impact of Sample Size on Accuracy Compute the standard error for sample proportions from a population with proportion \(p=0.4\) for sample sizes of \(n=30, n=200,\) and \(n=1000 .\) What effect does increasing the sample size have on the standard error? Using this information about the effect on the standard error, discuss the effect of increasing the sample size on the accuracy of using a sample proportion to estimate a population proportion.

In a survey of 1000 US adults, twenty percent say they never exercise. This is the highest level seen in five years. \({ }^{7}\) Find and interpret a \(99 \%\) confidence interval for the proportion of US adults who say they never exercise. What is the margin of error, with \(99 \%\) confidence?

A \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired data in the following table: $$ \begin{array}{lccccc} \hline \text { Case } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline \text { Treatment 1 } & 22 & 28 & 31 & 25 & 28 \\ \text { Treatment 2 } & 18 & 30 & 25 & 21 & 21 \\ \hline \end{array} $$

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether there is a difference between two groups in the proportion who voted, if 45 out of a random sample of 70 in Group 1 voted and 56 out of a random sample of 100 in Group 2 voted.

IQ tests scale the scores so that the mean IQ score is \(\mu=100\) and standard deviation is \(\sigma=15\). Suppose that 30 fourth graders in one class are given such an IQ test that is appropriate for their grade level. If the students are really a random sample of all fourth graders, what is the chance that the average IQ score for the class is above \(105 ?\)
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