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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblem 7.1.41a
Chapter 7, Problem 7.1.41a

No Failures According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)


a. If n independent trials result in no successes, why can’t we find confidence interval limits by using the methods described in this section?

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Step 1: Understand the Rule of Three. The Rule of Three is a statistical principle used to estimate the upper bound of the true population proportion when no successes are observed in a sample. It states that the upper bound of the proportion is approximately 3/n, where n is the sample size.
Step 2: Recognize the limitation of traditional confidence interval methods. Traditional methods for constructing confidence intervals rely on the presence of observed successes (x > 0) to calculate the sample proportion and its variability. When x = 0 (no successes), the sample proportion is zero, and the variability cannot be estimated using standard formulas.
Step 3: Consider the implications of x = 0. With no successes observed, the data provides no direct evidence of the true population proportion. This makes it impossible to calculate a meaningful lower bound for the confidence interval using traditional methods, as the lower bound would be zero or undefined.
Step 4: Understand why the Rule of Three is used instead. The Rule of Three provides a practical solution by focusing on the upper bound of the proportion. It assumes that the absence of observed successes is due to the rarity of the event, and it uses a conservative estimate (3/n) to ensure 95% confidence.
Step 5: Reflect on the context of the problem. The Rule of Three is particularly useful in situations where the event of interest is rare, and the sample size is relatively small. It provides a simple and robust way to estimate the upper bound of the population proportion without relying on traditional confidence interval methods.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In the context of proportions, it provides an estimate of the uncertainty around the sample proportion. When there are no successes in a sample, traditional methods for calculating confidence intervals may not apply, as they often rely on the presence of at least one success to establish a meaningful range.
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Introduction to Confidence Intervals

Rule of Three

The Rule of Three is a statistical principle used when there are zero successes in a sample. It states that if a sample of size n results in no successes, we can be 95% confident that the true proportion of successes in the population is less than or equal to 3/n. This rule provides a way to estimate an upper bound for the population proportion when traditional methods fail due to a lack of successes.
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Probability of Mutually Exclusive Events

Sample Size and Successes

Sample size (n) and the number of successes (x) are critical in determining the reliability of statistical estimates. In cases where n is small and x is zero, the lack of successes limits the ability to calculate a standard confidence interval, as the usual formulas require at least some successes to provide a valid estimate. Understanding the relationship between sample size and observed outcomes is essential for interpreting results in statistical analysis.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?


a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

Textbook Question

Analysis of Last Digits Weights of respondents were recorded as part of the California Health Interview Survey. The last digits of weights from 50 randomly selected respondents are listed below.



a. Use the bootstrap method with 1000 bootstrap samples to find a 95% confidence interval estimate of .

Textbook Question

Large Data Sets from Appendix B. In Exercises 21 and 22, use the data set in Appendix B. Assume that each sample is a simple random sample obtained from a population with a normal distribution.


Birth Weights Refer to Data Set 6 “Births” in Appendix B.


a. Use the 205 birth weights of girls to construct a 95% confidence interval estimate of the standard deviation of the population from which the sample was obtained.


Textbook Question

Freshman 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, , where represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

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a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the mean weight change for the population.


Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Job Interviews In a Harris poll of 514 human resource professionals, 90% said that the appearance of a job applicant is most important for a good first impression.


a. Among the 514 human resource professionals who were surveyed, how many of them said that the appearance of a job applicant is most important for a good first impression?


Textbook Question

15. HEIGHTS OF FEMALE SOCCER PLAYERS Listed below are the heights (in.) of players on the U.S. Women’s National Soccer Team (at the time of this writing). Use those heights as a sample of the heights of all professional women soccer players.

a. Use 1000 bootstrap samples to construct a 95% confidence interval estimate of σ.