Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.4.2

Chapter 6, Problem 6.4.2

What happens to the shape of the chi-square distribution as the degrees of freedom increase?

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Understand the chi-square distribution: The chi-square distribution is a continuous probability distribution that is commonly used in hypothesis testing and confidence interval estimation for variance. It is defined by its degrees of freedom (df).

Recall the relationship between degrees of freedom and the shape of the chi-square distribution: The degrees of freedom (df) determine the shape of the chi-square distribution. As df increases, the distribution changes its characteristics.

Note the behavior for small degrees of freedom: When the degrees of freedom are small (e.g., df = 1 or 2), the chi-square distribution is highly skewed to the right, meaning it has a long tail on the positive side.

Describe the effect of increasing degrees of freedom: As the degrees of freedom increase, the chi-square distribution becomes less skewed and starts to resemble a normal distribution. This is because the central limit theorem implies that the sum of a large number of independent squared standard normal variables approaches a normal distribution.

Summarize the trend: In summary, as the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches the shape of a normal distribution, with the peak shifting slightly to the right and the spread increasing.

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Key Concepts

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

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution that arises in statistics, particularly in hypothesis testing and confidence interval estimation for variance. It is defined by its degrees of freedom, which are typically related to the number of independent standard normal variables being squared and summed. The distribution is positively skewed, especially with low degrees of freedom.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of the chi-square distribution, the degrees of freedom are often determined by the number of categories minus one. As the degrees of freedom increase, the distribution becomes less skewed and approaches a normal distribution.

Shape of the Distribution

The shape of a distribution describes how the values are spread out across different ranges. For the chi-square distribution, as the degrees of freedom increase, the distribution becomes more symmetric and approaches a bell-shaped curve. This transition indicates that with more data, the variability in the sample estimates becomes more stable, leading to a distribution that resembles the normal distribution.

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Textbook Question

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

Textbook Question

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

In a survey of 2094 U.S. adults who have used an online dating app, 57% said their personal experience with online dating was positive. The survey’s margin of error is ±3.6%. (Source: Pew Research Center)

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.

c = 0.99, n = 30

Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(21.61, 30.15)

Textbook Question

When estimating the population mean, why not construct a 99% confidence interval every time?

Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.