91Ó°ÊÓ

The number of hours a lightbulb burns before failing varies from bulb to bulb. The population distribution of burnout times is strongly skewed to the right. The central limit theorem says that (a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean \(\mu\) for all bulbs of this type. (b) the average burnout time of a large number of bulbs has a sampling distribution with the same shape (strongly skewed) as the population distribution. (c) the average burnout time of a large number of bulbs has a sampling distribution with similar shape but not as extreme (skewed, but not as strongly) as the population distribution. (d) the average burnout time of a large number of bulbs has a sampling distribution that is close to Normal. (e) the average burnout time of a large number of bulbs has a sampling distribution that is exactly Normal.

Step by step solution

01

Understanding the Central Limit Theorem

The central limit theorem states that the sampling distribution of the sample mean will tend to be Normal or close to Normal if the sample size is large enough, regardless of the shape of the population distribution.

02

Analyzing Given Options

Option (a) talks about the average getting closer to the mean, which is true but does not describe the distribution. Option (b) suggests the sampling distribution has the same skewed shape as the population, which is incorrect. Option (c) states that the shape of the distribution is similar but less extreme, which is incorrect for large samples. Option (d) indicates the sampling distribution is close to Normal for large samples, which aligns with the central limit theorem. Option (e) suggests the distribution is exactly Normal, which is a simplification and not generally accurate.

03

Applying Central Limit Theorem Conclusion

Given the choices, option (d) correctly describes that the sampling distribution of the average burnout time of a large number of bulbs is close to Normal due to the central limit theorem.

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Key Concepts

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

Sampling Distribution

When discussing the Central Limit Theorem, one key idea is the concept of a sampling distribution. The sampling distribution is essentially the probability distribution of a statistic, like the sample mean, obtained from a large number of samples drawn from a specific population. As more data is collected, the average or mean of the sample statistics begins to form a predictable pattern.

Key points about sampling distributions include:

• The mean of the sampling distribution is typically equal to the mean of the underlying population.
• With larger sample sizes, the sampling distribution tends to be more Normal, even if the population distribution is skewed.
• The standard deviation of the sampling distribution, known as the standard error, decreases as sample size increases, indicating more precise estimations of the population parameter.

This concept is critical in statistical analysis as it ensures that inferences about populations can be made from samples in a reliable manner, especially when large samples are involved.

Population Distribution

The population distribution represents how a particular variable, like the burnout time of lightbulbs, is distributed across all items in the population. While a population distribution can take on many shapes, it often has characteristics such as being skewed or symmetric.

Key characteristics of population distributions include:

• They provide a complete overview of data across an entire population, giving insights into overall trends and tendencies.
• They can be skewed, meaning they may not always be symmetric. For the lightbulb example, the distribution is skewed to the right.
• Population distributions contain all the data points, while sample distributions contain only a subset.

Understanding population distribution is essential because it helps contextualize sample data and interpret how the sampled data’s distribution may differ from the overall population.

Normal Distribution

The Normal distribution, often called the Gaussian distribution, is a fundamental concept in statistics due to its properties and widespread occurrence in nature. It is famously known for its bell-shaped appearance, with the bulk of the data peaking around the mean.

Features of a Normal distribution include:

• Symmetry about the mean, implying that each half of the distribution is a mirror image.
• A predictable pattern where approximately 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three.
• The mean, median, and mode of a Normal distribution are all equal, located at the center of the distribution.

The importance of Normal distribution is magnified by the Central Limit Theorem, which states that with a sufficiently large sample size, the sample means will be approximately Normally distributed, regardless of the population distribution’s shape. This property is particularly significant in making statistical inferences about populations based on sample data, as many statistical methods rely on the Normality assumption.