/*! 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 26 In anthropological studies, an i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 26

In anthropological studies, an important characteristic of fossils is cranial capacity. Frequently skulls are at least partially decomposed, so it is necessary to use other characteristics to obtain information about capacity. One such measure that has been used is the length of the lambda-opisthion chord. The article "Vertesszollos and the Presapiens Theory" (American Journal of Physical Anthropology \([1971]\) ) reported the accompanying data for \(n=7\) Homo erectus fossils. \(x\) (chord \(\begin{aligned}&\text { length in } \mathrm{mm} \text { ) } & 78 & 75 & 78 & 81 & 84 & 86 & 87\end{aligned}\) \(y\) (capacity in \(\mathrm{cm}^{3}\) ) \(\begin{array}{lllllll}850 & 775 & 750 & 975 & 915 & 1015 & 1030\end{array}\) Suppose that from previous evidence, anthropologists had believed that for each 1 -mm increase in chord length, cranial capacity would be expected to increase by \(20 \mathrm{~cm}^{3}\). Do these new experimental data strongly contradict prior belief?

Short Answer

Expert verified
To provide a short answer, one must complete the detailed steps in order to arrive at the actual slope (change in cm鲁 per mm). If the slope matches closely with the expected change as per the previous belief of anthropologists, i.e., 20 cm鲁 per mm, the new experimental data can be considered supportive. However, if there is a considerable deviation, the data can be seen as contradicting the prior belief.

Step by step solution

01

Calculate the averages

The averages of both chord length (\( x \)) and cranial capacity (\( y \)) first need to be calculated based on the provided data. This involves adding up all the individual data points for \( x \) and \( y \) respectively, and then dividing each total by the number of data points \( n = 7 \).
02

Compute the slope of the regression line

The slope of the regression line, also known as the rate of change, is estimated as it gives the predicted change in cranial capacity (\( y \)) per unit increase in chord length (\( x \)). The formula to calculate the slope is as follows: \( \beta = \frac{{\Sigma(x_i-\overline{x})(y_i-\overline{y})}}{{\Sigma(x_i-\overline{x})^2}} \), where \( \overline{x} \) and \( \overline{y} \) are the average of \( x \) and \( y \) values respectively, \( x_i \) and \( y_i \) are individual \( x \) and \( y \) data points. To find the slope, this exercise requires to first determine the numerator \( \Sigma(x_i-\overline{x})(y_i-\overline{y}) \) and denominator \( \Sigma(x_i-\overline{x})^2 \), and then divide the numerator by the denominator.
03

Assess the resultant slope

Now, compare the slope obtained from step 2 with the initially held belief of the anthropologists, which is \( 20 \, cm^3/mm \). This can be done by looking at the magnitude and direction (positive/negative) of the slope. If the calculated slope greatly deviates from \( 20 \, cm^3/mm \), this suggests that the data contradicts the prior belief held by the anthropologists.

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.

Anthropological Studies
Anthropological studies delve into the rich history of human evolution, often piecing together life's story from the remains left behind. One such puzzle piece is cranial capacity, which can reveal important insights into the neurological and biological development of early human ancestors like Homo erectus. Cranial capacity, or the volume of the braincase, is correlated with various other skeletal measurements to infer cognitive abilities and social behavior. However, due to the fragmentary nature of fossil records, anthropologists must frequently rely on proxy measures, such as the length of the lambda-opisthion chord, to estimate cranial capacity. Such indirect methods underscore the importance of statistical tools to validate assumptions about evolutionary biology and the relationships between different physical characteristics.

Through meticulous measurements of fossil remains and subsequent statistical analysis, researchers can draw conclusions about the evolution of brain size and its implications for human advancement. This fascinating intersection of archeology, biology, and statistics facilitates a deeper understanding of our ancestral heritage.
Homo Erectus Fossils
The Homo erectus, a significant chapter in the human evolutionary saga, roamed the earth approximately 1.8 million to 30,000 years ago. Fossils of this species provide critical insights into the development of human traits such as increased brain size and the advent of bipedalism. The lambda-opisthion chord, often used in anthropological research, is a measurement from the skull's lambda point to the opisthion. This measurement provides an estimate for the cranial capacity when the skull is partly decomposed and direct volume measurements are impossible.

Investigating Homo erectus fossils involves comparing cranial capacities to understand changes across different populations and time frames. Studying these variations helps in reconstructing evolutionary patterns and in hypothesizing about the behaviors and lifestyles of these ancient beings.
Regression Line Slope
In the context of regression analysis, the slope of the regression line is a crucial element, representing the expected change in the dependent variable for each unit increase in the independent variable. For anthropologists examining cranial capacity, this means looking at how much an increase in the lambda-opisthion chord length can predict an increase in brain volume.

Calculating the regression line slope involves statistical methods that blend the independent values (lambda-opisthion chord lengths) against the dependent values (cranial capacities). If the average increase in cranial capacity substantially contrasts the slope derived from the regression analysis, it can signal the need to adjust current theories or models. The slope is a testament to the evolutionary theories proposed by anthropologists and stands as a quantitative testament to their research.
Statistical Hypothesis Testing
Statistical hypothesis testing is a rigorous method used by scientists, including anthropologists, to test the validity of their assumptions against empirical data. This process involves stating a null hypothesis, which typically reflects the current belief or theory, and an alternative hypothesis that represents a new claim or belief. In examining Homo erectus fossils, a hypothesis test could be constructed to determine whether the actual increase in cranial capacity per millimeter increase in chord length differs from the expected 20 cm3/mm.

To test this hypothesis, researchers calculate the regression line slope and then use statistical tests, such as t-tests, to determine if there's a significant difference between the observed slope and the one hypothesized. The outcome helps scientists decide whether to uphold their current beliefs or consider new evidence that could reshape understanding of human evolution.

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

'The article "Photocharge Effects in Dye Sensitized \(\mathrm{Ag}[\mathrm{Br}, \mathrm{I}]\) Emulsions at Millisecond Range Exposures" (Photographic Science and Engineering [1981]: \(138-144\) ) gave the accompanying data on \(x=\%\) light absorption and \(y=\) peak photovoltage. \(\begin{array}{rrrrrrrrrr}x & 4.0 & 8.7 & 12.7 & 19.1 & 21.4 & 24.6 & 28.9 & 29.8 & 30.5 \\ y & 0.12 & 0.28 & 0.55 & 0.68 & 0.85 & 1.02 & 1.15 & 1.34 & 1.29\end{array}\) \(\sum x=179.7 \quad \sum x^{2}=4334.41\) \(\sum y=7.28 \quad \sum y^{2}=7.4028 \quad \sum x y=178.683\) a. Construct a scatterplot of the data. What does it suggest? b. Assuming that the simple linear regression model is appropriate, obtain the equation of the estimated regression line. c. How much of the observed variation in peak photovoltage can be explained by the model relationship? d. Predict peak photovoltage when percent absorption is 19.1, and compute the value of the corresponding residual. e. The authors claimed that there is a useful linear relationship between the two variables. Do you agree? Carry out a formal test. f. Give an estimate of the average change in peak photovoltage associated with a \(1 \%\) increase in light absorption. Your estimate should convey information about the precision of estimation. g. Give an estimate of true average peak photovoltage when percentage of light absorption is 20 , and do so in a way that conveys information about precision.

The accompanying data on \(x=\) treadmill run time to exhaustion (min) and \(y=20-\mathrm{km}\) ski time (min) were taken from the article "Physiological Characteristics and Performance of Top U.S. Biathletes" (Medicine and \(S c i\) ence in Sports and Exercise \([1995]: 1302-1310):\) \(\begin{array}{lrrrrrr}x & 7.7 & 8.4 & 8.7 & 9.0 & 9.6 & 9.6 \\ y & 71.0 & 71.4 & 65.0 & 68.7 & 64.4 & 69.4 \\ x & 10.0 & 10.2 & 10.4 & 11.0 & 11.7 & \\ y & 63.0 & 64.6 & 66.9 & 62.6 & 61.7 & \\ \sum x & =106.3 & \sum x^{2}=1040.95 & & \\ \sum y= & 728.70 & \sum x y=7009.91 & \sum y^{2}=48390.79\end{array}\) a. Does a scatterplot suggest that the simple linear regression model is appropriate? b. Determine the equation of the estimated regression line, and draw the line on your scatterplot. c. What is your estimate of the average change in ski time associated with a 1 -min increase in treadmill time? d. What would you predict ski time to be for an individual whose treadmill time is \(10 \mathrm{~min} ?\) e. Should the model be used as a basis for predicting ski time when treadmill time is \(15 \mathrm{~min}\) ? Explain. f. Calculate and interpret the value of \(r^{2}\). g. Calculate and interpret the value of \(s_{e}\)

The employee relations manager of a large company was concemed that raises given to employees during a recent period might not have been based strictly on objective performance criteria. A sample of \(n=20\) employees was selected, and the values of \(x\), a quantitative measure of productivity, and \(y\), the percentage salary increase, were determined for each one. A computer package was used to fit the simple linear regression model, and the resulting output gave the \(P\) -value \(=.0076\) for the model utility test. Does the percentage raise appear to be linearly related to productivity? Explain.

Let \(x\) be the size of a house (sq \(\mathrm{ft}\) ) and \(y\) be the amoun of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq ft of space? d. What is the average change in usage associated with a 1 -sq-ft increase in size? e. What is the average change in usage associated with a 100 -sq-ft increase in size? f. Would you use the model to predict mean usage for a 500 -sq-ft house? Why or why not? (Note: There are no small houses in the community in which this model is valid.)

The data of Exercise \(13.25\), in which \(x=\) milk temperature and \(y=\) milk \(\mathrm{pH}\), yield $$ \begin{aligned} &n=16 \quad \bar{x}=43.375 \quad S_{x x}=7325.75 \\ &b=-.00730608 \quad a=6.843345 \quad s_{e}=.0356 \end{aligned} $$ a. Obtain a \(95 \%\) confidence interval for \(\alpha+\beta(40)\), the true average milk \(\mathrm{pH}\) when the milk temperature is \(40^{\circ} \mathrm{C}\). b. Calculate a \(99 \%\) confidence interval for the true average milk \(\mathrm{pH}\) when the milk temperature is \(35^{\circ} \mathrm{C}\). c. Would you recommend using the data to calculate a \(95 \%\) confidence interval for the true average \(\mathrm{pH}\) when the temperature is \(90^{\circ} \mathrm{C} ?\) Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.