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Chapter 9: Problem 5

True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.

Short Answer

Expert verified
Typically true, but sample size matters.

Step by step solution

01

- Understand the Concept of Confidence Interval

A confidence interval is used to estimate the population parameter (such as the mean) based on a sample statistic. It provides a range of values that is likely to contain the population parameter.
02

- Normality of the Population Distribution

For constructing a confidence interval about the mean, the requirement of the normality of the population distribution depends on the sample size. According to the Central Limit Theorem, for large samples (typically n > 30), the sampling distribution of the sample mean will be approximately normal regardless of the population distribution.
03

- Small Sample Sizes

When the sample size is small (n 鈮� 30), the population from which the sample is drawn should be approximately normal to accurately use the t-distribution for constructing the confidence interval.
04

- Conclusion

The statement can be considered as generally true but with the important consideration of sample size. Therefore, the answer is not a straightforward true or false without the context of the sample size.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the sampling distribution of the sample mean will be approximately normal for a sufficiently large sample size, regardless of the population's distribution shape.

This theorem comes into play when we construct confidence intervals. Even if the original population is skewed or not normally distributed, the sample means will form a normal distribution as the sample size increases.

Typically, a sample size greater than 30 is considered large enough for the CLT to hold true. This means that even with non-normal populations, we can use methods that assume normality once we have a large sample.
Sample Size
Sample size significantly impacts the accuracy of a confidence interval. A larger sample size provides a more precise estimate of the population parameter, reducing the margin of error.

When the sample size is small (usually n 鈮� 30), the distribution of the sample mean may not be well-approximated by a normal distribution. For small samples, the underlying population distribution needs to be approximately normal to make accurate inferences.

In general, a larger sample size:
  • Reduces the variability in the sampling distribution
  • Makes the confidence interval narrower
  • Improves the reliability of the results
t-distribution
The t-distribution is used when constructing confidence intervals for small sample sizes. It looks similar to the normal distribution but has thicker tails. These thicker tails account for the additional variability that is expected when estimating parameters from small samples.

The t-distribution is characterized by degrees of freedom, typically calculated as the sample size minus one (n - 1). As the sample size increases, the t-distribution approaches the normal distribution.

Here's why the t-distribution is important:
  • It adjusts for the smaller sample size
  • Accounts for the additional uncertainty
  • Works with sample means to provide accurate confidence intervals
Normality Assumption
The normality assumption is crucial when dealing with small samples and applying the t-distribution. This assumption means that the data from the population should follow a normal distribution.

When the normality assumption holds, the resulting confidence intervals will be accurate and reliable. If the population distribution is not normal and the sample size is small, the confidence interval estimates might be invalid.

Ways to check for normality include:
  • Visual inspections like Q-Q plots
  • Statistical tests such as the Shapiro-Wilk test
Understanding these methods and their importance helps ensure that the results from your statistical analysis are trustworthy.

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Most popular questions from this chapter

A simple random sample of size \(n\) is drawn from a population that is known to be normally distributed. The sample variance, \(s^{2}\) is determined to be 12.6 . (a) Construct a \(90 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n,\) is 20 (b) Construct a \(90 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is \(30 .\) How does increasing the sample size affect the width of the interval? (c) Construct a \(98 \%\) confidence interval for \(\sigma^{2}\) if the sample size, \(n\), is 20. Compare the results with those obtained in part (a). How does increasing the level of confidence affect the width of the confidence interval?

An urban economist wishes to estimate the proportion of Americans who own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with \(90 \%\) confidence if (a) he uses a 2010 estimate of 0.669 obtained from the U.S. Census Bureau? (b) he does not use any prior estimates?

How much time do Americans spend eating or drinking? Suppose for a random sample of 1001 Americans age 15 or older, the mean amount of time spent eating or drinking per day is 1.22 hours with a standard deviation of 0.65 hour. Source: American Time Use Survey conducted by the Bureau of Labor Statistics (a) A histogram of time spent eating and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day. (b) There are over 200 million Americans age 15 or older. Explain why this, along with the fact that the data were obtained using a random sample, satisfies the requirements for constructing a confidence interval. (c) Determine and interpret a \(95 \%\) confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day. (d) Could the interval be used to estimate the mean amount of time a 9-year- old American spends eating and drinking each day? Explain.

True or False: The chi-square distribution is symmetric.

A simple random sample of size \(n\) is drawn from a population that is normally distributed. The sample mean, \(\bar{x},\) is found to be \(50,\) and the sample standard deviation, \(s,\) is found to be \(8 .\) (a) Construct a \(98 \%\) confidence interval for \(\mu\) if the sample size, \(n,\) is 20 (b) Construct a \(98 \%\) confidence interval for \(\mu\) if the sample size, \(n\), is \(15 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(95 \%\) confidence interval for \(\mu\) if the sample size, \(n\), is 20. Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, \(E\) ? (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?
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