Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.35

Chapter 6, Problem 6.3.35

Why Check It? Why is it necessary to check that np^ ≥ 5 and nq^ ≥ 5?

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The condition np̂ ≥ 5 and nq̂ ≥ 5 is used to ensure that the sampling distribution of the sample proportion p̂ is approximately normal. This is important because many statistical methods, such as hypothesis testing and confidence intervals, rely on the assumption of normality.

Here, n represents the sample size, p̂ is the sample proportion, and q̂ = 1 - p̂ is the complement of the sample proportion. These conditions ensure that there are enough successes (np̂) and failures (nq̂) in the sample to approximate a normal distribution.

When np̂ and nq̂ are both at least 5, the Central Limit Theorem applies, which states that the sampling distribution of p̂ will be approximately normal, regardless of the shape of the population distribution.

If these conditions are not met (i.e., np̂ < 5 or nq̂ < 5), the sampling distribution may be skewed or not well-approximated by a normal distribution, leading to inaccurate results in statistical inference.

To summarize, checking these conditions ensures the validity of using normal approximation methods for proportions, which is a critical step in solving problems involving proportions in statistics.

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Key Concepts

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

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success). Understanding this distribution is crucial for determining when certain statistical methods can be applied.

Normal Approximation

The normal approximation to the binomial distribution allows us to use the normal distribution to estimate probabilities for binomial outcomes when certain conditions are met. Specifically, the conditions np ≥ 5 and nq ≥ 5 ensure that the distribution is sufficiently symmetric and bell-shaped, making the approximation valid.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the original distribution of the data. This theorem underpins the rationale for checking the conditions np ≥ 5 and nq ≥ 5, as it guarantees that the sampling distribution will be approximately normal under these conditions.

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Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

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Textbook Question

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Textbook Question

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112

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Textbook Question

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205

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Textbook Question

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