Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problema 6.3.35

Capitolo 6, Problema 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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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.

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03:28

Mean & Standard Deviation of Binomial Distribution

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.

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Using the Normal Distribution to Approximate Binomial Probabilities

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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04:52

Calculating the Mean

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Domanda del libro di testo

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

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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.

c = 0.99, n = 15

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In Exercise 28, the population mean weekly time spent on homework by students is 7.8 hours. Does the t-value fall between -t0.99 and t0.99?

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In Exercises 15 and 16, find the t-value for the given values of xbar, μ, s and n.

xbar = 70.3, μ = 64.8, s = 7.1, n = 16

112

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Describe how the t-distribution curve changes as the sample size increases.

205

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Constructing Confidence Intervals In Exercises 13 and 14, construct a 99% confidence interval for the population proportion. Interpret the results.

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