/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 True or False: To construct a co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with \(98 \%\) confidence if (a) he uses a 2006 estimate of \(15 \%\) obtained from a Coinstar National Currency Poll? (b) he does not use any prior estimate?

A simple random sample of size \(n<30\) for \(a\) quantitative variable has been obtained. Using the normal probability plot, the correlation between the variable and expected z-score, and the boxplot, judge whether a t-interval should be constructed. $$ n=9 ; \text { Correlation }=0.997 $$

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?

The trade volume of a stock is the number of shares traded on a given day. The following data, in millions (so that 6.16 represents 6,160,000 shares traded), represent the volume of PepsiCo stock traded for a random sample of 40 trading days in 2014. \begin{array}{llllllll} \hline 6.16 & 6.39 & 5.05 & 4.41 & 4.16 & 4.00 & 2.37 & 7.71 \\ \hline 4.98 & 4.02 & 4.95 & 4.97 & 7.54 & 6.22 & 4.84 & 7.29 \\ \hline 5.55 & 4.35 & 4.42 & 5.07 & 8.88 & 4.64 & 4.13 & 3.94 \\ \hline 4.28 & 6.69 & 3.25 & 4.80 & 7.56 & 6.96 & 6.67 & 5.04 \\ \hline 7.28 & 5.32 & 4.92 & 6.92 & 6.10 & 6.71 & 6.23 & 2.42 \\ \hline \end{array} (a) Use the data to compute a point estimate for the population mean number of shares traded per day in 2014 (b) Construct a \(95 \%\) confidence interval for the population mean number of shares traded per day in 2014 . Interpret the confidence interval. (c) A second random sample of 40 days in 2014 resulted in the data shown next. Construct another \(95 \%\) confidence interval for the population mean number of shares traded per day in 2014\. Interpret the confidence interval. $$ \begin{array}{llllrlll} \hline 6.12 & 5.73 & 6.85 & 5.00 & 4.89 & 3.79 & 5.75 & 6.04 \\ \hline 4.49 & 6.34 & 5.90 & 5.44 & 10.96 & 4.54 & 5.46 & 6.58 \\ \hline 8.57 & 3.65 & 4.52 & 7.76 & 5.27 & 4.85 & 4.81 & 6.74 \\ \hline 3.65 & 4.80 & 3.39 & 5.99 & 7.65 & 8.13 & 6.69 & 4.37 \\ \hline 6.89 & 5.08 & 8.37 & 5.68 & 4.96 & 5.14 & 7.84 & 3.71 \\ \hline \end{array} $$ (d) Explain why the confidence intervals obtained in parts (b) and (c) are different.

Indicate whether a confidence interval for a proportion or mean should be constructed to estimate the variable of interest. Justify your response. A developmental mathematics instructor wishes to estimate the typical amount of time students dedicate to studying mathematics in a week. She asks a random sample of 50 students enrolled in developmental mathematics at her school to report the amount of time spent studying mathematics in the past week.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.