/*! 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 1 The 20 brain volumes \(\left(\ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 1

The 20 brain volumes \(\left(\mathrm{cm}^{3}\right)\) from Data Set 8 "IQ and Brain Size" in Appendix B vary from a low of \(963 \mathrm{cm}^{3}\) to a high of \(1439 \mathrm{cm}^{3} .\) Use the range rule of thumb to estimate the standard deviation \(s\) and compare the result to the exact standard deviation of \(124.9 \mathrm{cm}^{3}\).

Short Answer

Expert verified
The estimated standard deviation is 119 cm鲁, and the exact value is 124.9 cm鲁.

Step by step solution

01

- Identify the Range

Find the range of the brain volumes by subtracting the smallest value from the largest value. In this case, the largest value is 1439 cm鲁 and the smallest value is 963 cm鲁. So, the range is: \( \text{Range} = 1439 \text{ cm}^3 - 963 \text{ cm}^3 = 476 \text{ cm}^3 \)
02

- Apply the Range Rule of Thumb

According to the range rule of thumb, the standard deviation (s) can be approximated by dividing the range by 4. Thus, \( s \text{ (estimated)} = \frac{ \text{Range} } { 4 } = \frac{ 476 \text{ cm}^3 } { 4 } = 119 \text{ cm}^3 \)
03

- Compare with the Exact Standard Deviation

The exact standard deviation given is 124.9 cm鲁. Comparing this to the estimated standard deviation: \( s \text{ (exact)} = 124.9 \text{ cm}^3 \) Thus, the estimated standard deviation is 119 cm鲁, which is very close to the exact value of 124.9 cm鲁.

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.

range rule of thumb
The range rule of thumb is a quick method for estimating the standard deviation of a dataset. It involves using the range, which is the difference between the maximum and minimum values in the dataset.

To apply this rule, you follow these simple steps:
  • Calculate the range: Subtract the smallest value from the largest value in your dataset.
  • Divide the range by 4: This gives you an estimated standard deviation.

For example, if the largest brain volume is 1439 cm鲁 and the smallest is 963 cm鲁, the range is 476 cm鲁.

Dividing this range by 4 gives an estimated standard deviation of 119 cm鲁. This estimation provides a quick way to understand variability without detailed calculations.
brain volumes
Brain volume measurements can vary significantly among individuals, reflecting differences in brain size due to numerous factors such as age, sex, and overall health.

In the given exercise, we have brain volumes ranging from 963 cm鲁 to 1439 cm鲁. These measurements help illustrate the variability that standard deviation aims to quantify.

Understanding brain volumes is crucial for various fields, including neuroscience and psychology. It allows researchers to study correlations between brain size and cognitive function, and to identify any abnormalities or changes over time.
standard deviation comparison
Once you have an estimated standard deviation using the range rule of thumb, it's useful to compare it with the exact standard deviation.

The exact standard deviation is calculated using all data points in the dataset, providing a more precise measure of variability.

In our example, the estimated standard deviation was 119 cm鲁, while the exact standard deviation was 124.9 cm鲁.

  • This comparison shows that the range rule of thumb provided a close approximation.
  • Such comparisons are important to understand the reliability of quick estimation methods like the range rule of thumb.

Having a close estimate helps save time and provides a quick sense of data spread, but for exact analyses, the precise calculation should be used.

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

Use z scores to compare the given values. Oscars In the 87 th Academy Awards, Eddie Redmayne won for best actor at the age of 33 and Julianne Moore won for best actress at the age of \(54 .\) For all best actors, the mean age is 44.1 years and the standard deviation is 8.9 years. For all best actresses, the mean age is 36.2 years and the standard deviation is 11.5 years. (All ages are determined at the time of the awards ceremony.) Relative to their genders, who had the more extreme age when winning the Oscar: Eddie Redmayne or Julianne Moore? Explain.

Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.) a. Find the variance \(\sigma^{2}\) of the population \(\\{9 \text { cigarettes, } 10 \text { cigarettes, } 20 \text { cigarettes }\\}\) b. After listing the nine different possible samples of two values selected with replacement, find the sample variance \(s^{2}\) (which includes division by \(n-1\) ) for each of them; then find the mean of the nine sample variances \(s^{2}\) c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by \(n\) ); then find the mean of those nine population variances. d. Which approach results in values that are better estimates of \(\sigma^{2}:\) part (b) or part (c)? Why? When computing variances of samples, should you use division by \(n\) or \(n-1 ?\) e. The preceding parts show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\). Is \(s\) an unbiased estimator of \(\sigma ?\) Explain.

Find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44. I years; (Exercise 31) 224.3; (Exercise 32) 255.I. $$\begin{array}{|c|c|}\hline \begin{array}{c}\text { Age (yr) of Best Actress } \\\\\text { When Oscar Was Won }\end{array} & \text { Frequency } \\\\\hline 20-29 & 29 \\\\\hline 30-39 & 34 \\ \hline 40-49 & 14 \\\\\hline 50-59 & 3 \\\\\hline 60-69 & 5 \\\\\hline 70-79 & 1 \\\\\hline 80-89 & 1 \\\\\hline\end{array}$$

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-I, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below in dollars are the annual costs of tuition and fees at the 10 most expensive colleges in the United States for a recent year (based on data from \(U . S .\) News \& World Report). The colleges listed in order are Columbia, Vassar, Trinity, George Washington, Carnegie Mellon, Wesleyan, Tulane, Bucknell, Oberlin, and Union. What does this "top 10" list tell us about the variation among costs for the population of all U.S. college tuition? $$\begin{array}{rlllllllll} 49,138 & 47,890 & 47,510 & 47,343 & 46,962 & 46,944 & 46,930 & 46,902 & 46,870 & 46,785 \end{array}$$

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype codes, where \(1=\) smooth-yellow, \(2=\) smooth-green, \(3=\) wrinkled- yellow, and \(4=\) wrinkled-green. Can the measures of center be obtained for these values? Do the results make sense? 2 \(\begin{array}{rrrrrrrrrrrrr}1 & 4 & 1 & 2 & 2 & 1 & 2 & 3 & 3 & 2 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 2 & 2\end{array}\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.