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Chapter 8: Problem 106

Need 15 successes and 15 failures To use the largesample confidence interval for \(p,\) you need at least 15 successes and 15 failures. Show that the smallest value of \(n\) for which the method can be used is (a) 30 when \(\hat{p}=0.50,\) (b) 50 when \(\hat{p}=0.30,\) (c) 150 when \(\hat{p}=0.10\). That is, the overall \(n\) must increase as \(\hat{p}\) moves toward 0 or 1 . (When the true proportion is near 0 or \(1,\) the sampling distribution can be highly skewed unless \(n\) is quite large.)

Short Answer

Expert verified
The smallest \( n \) required is 30 for \( \hat{p}=0.50 \), 50 for \( \hat{p}=0.30 \), and 150 for \( \hat{p}=0.10 \).

Step by step solution

01

Understand the Condition

We need at least 15 successes and 15 failures to use the large-sample confidence interval method for the proportion \( p \). This means \( np \geq 15 \) and \( n(1-p) \geq 15 \).
02

Set Up Equations for \(\hat{p} = 0.50 \)

For \( \hat{p} = 0.50 \), the equations become \( n(0.50) \geq 15 \) and \( n(1-0.50) \geq 15 \), which simplifies to \( 0.50n \geq 15 \).
03

Solve for Minimum \( n \) when \( \hat{p} = 0.50 \)

Solving \( 0.50n \geq 15 \) gives \( n \geq 30 \). Thus, the smallest \( n \) is 30.
04

Set Up Equations for \( \hat{p} = 0.30 \)

For \( \hat{p} = 0.30 \), the conditions are \( n(0.30) \geq 15 \) and \( n(0.70) \geq 15 \).
05

Solve for Minimum \( n \) when \( \hat{p} = 0.30 \)

Solving these, we find \( 0.30n \geq 15 \) gives \( n \geq 50 \) and \( 0.70n \geq 15 \) confirms \( n \geq 21.43 \). Thus, the smallest \( n \) is 50.
06

Set Up Equations for \( \hat{p} = 0.10 \)

For \( \hat{p} = 0.10 \), the conditions are \( n(0.10) \geq 15 \) and \( n(0.90) \geq 15 \).
07

Solve for Minimum \( n \) when \( \hat{p} = 0.10 \)

Solving \( 0.10n \geq 15 \) gives \( n \geq 150 \), and verifying \( 0.90n \geq 15 \) gives \( n \geq 16.67 \). Thus, the smallest \( n \) is 150.
08

Conclusion on \( n \) and \( \hat{p} \) Relationship

As \( \hat{p} \) moves further from 0.5 towards 0 or 1, a larger sample size \( n \) is required to maintain at least 15 successes and failures. The distribution becomes skewed, requiring more data for accuracy.

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Key Concepts

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

Sample Size
Sample size, represented by \( n \), is the number of individual data points or observations collected in a study. A sufficient sample size is crucial for the accuracy of statistical analyses, such as calculating confidence intervals for a proportion. In the context of proportion confidence intervals, the sample size must be large enough for statistical assumptions to hold. This means that both the number of successes \( np \) and the number of failures \( n(1-p) \) should be at least 15.
This ensures that the sampling distribution of the proportion is approximately normal, which is a requirement for using the large-sample confidence interval method.
Proportion
Proportion refers to a part or fraction of a whole, often represented as \( p \). In statistics, it reflects the probability of a particular outcome occurring. For instance, if you're analyzing a dataset to find the proportion of students who passed an exam, that will be \( p \).
In confidence interval calculations, the sample proportion \( \hat{p} \) is used to estimate the true population proportion. It's crucial that \( \hat{p} \) is based on a sample where both events we are interested in (successes and failures) occur frequently enough. As shown in the exercise, if \( \hat{p} \) approaches values close to 0 or 1, ensuring a minimum sample size gets more challenging.
Statistical Distribution
Statistical distribution describes how values in a dataset are spread out. For proportions, the sampling distribution is approximately normal if the sample size conditions are satisfied: at least 15 successes and 15 failures. This normality is what allows statisticians to use the large-sample confidence interval for \( p \).
A normal distribution is symmetrical, meaning the data cascades evenly around the mean, which isn't the case if there are too few successes or failures, leading to skewed distributions. As \( \hat{p} \) becomes extreme (near 0 or 1), the distribution skews more, necessitating larger samples to retain accuracy in statistical analysis.
Successes and Failures
In statistical terms, 'successes' and 'failures' refer to the two possible outcomes in a binary dataset. For instance, in a survey, responses may be classified as 'yes' (success) or 'no' (failure).
The requirement for at least 15 successes and 15 failures for confidence interval calculation ensures the sample proportion is accurately estimated. This condition helps maintain the normal approximation's validity, allowing reliable and meaningful confidence intervals. Therefore, when \( \hat{p} \) is skewed towards 0 or 1, ensuring that both outcome types are sufficiently represented in the data is vital for valid conclusions.

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Most popular questions from this chapter

Vegetarianism Time magazine (July 15,2002 ) quoted a poll of 10,000 Americans in which only \(4 \%\) said they were vegetarians. a. What has to be assumed about this sample to construct a confidence interval for the population proportion of vegetarians? b. Construct a \(99 \%\) confidence interval for the population proportion. Explain why the interval is so narrow, even though the confidence level is high. c. In interpreting this confidence interval, can you conclude that fewer than \(10 \%\) of Americans are vegetarians? Explain your reasoning.

Born again A poll of a random sample of \(n=2000\) Americans by the Pew Research Center (www.peoplepress.org) indicated that \(36 \%\) considered themselves "born-again" or evangelical Christians. How would you explain to someone who has not studied statistics: a. What it means to call this a point estimate. b. Why this does not mean that exactly \(36 \%\) of all Americans consider themselves to be born-again or evangelical Christians.

Religious beliefs A column by New York Times columnist Nicholas Kristof (August 15,2003 ) discussed results of polls indicating that religious beliefs in the United States tend to be quite different from those in other Western nations. He quoted recent Gallup and Harris polls of random samples of about 1000 Americans estimating that \(83 \%\) believe using the Virgin Birth of Jesus but only \(28 \%\) believe in evolution. A friend of yours is skeptical, claiming that it's impossible to predict beliefs of over 200 million adult Americans by interviewing only 1000 of them. Write a one-page report using this context to show how you could explain about random sampling, the margin of error, and how a margin of error depends on the sample size.

Why bootstrap? Explain the purpose of using the bootstrap method.

Exit poll predictions A national television network takes an exit poll of 1400 voters after each has cast a vote in a state gubernatorial election. Of them, 660 say they voted for the Democratic candidate and 740 say they voted for the Republican candidate. a. Treating the sample as a random sample from the population of all voters, would you predict the winner? Base your decision on a \(95 \%\) confidence interval. b. Base your decision on a \(99 \%\) confidence interval. Explain why you need stronger evidence to make a prediction when you want greater confidence.
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