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• Chair, of the Department of Statistics https://publichealth.berkeley.edu/people/sandrine-dudoit/ The following table shows scores on the first quiz (maximum score 10 points) for eighth-grade students in an introductory level French course arrhythmia nos order 75 mg triamterene with visa. The instructor grouped the students in the course as follows: Group 1: Never studied foreign language before but have good English skills Group 2: Never studied foreign language before and have poor English skills Group 3: Studied at least one other foreign language 14 blood pressure medication zanidip 75mg triamterene visa. Both observations used a rating scale of 0­10 arrhythmias in children order triamterene cheap, with 0 = very poor and 10 = excellent hypertension knowledge questionnaire order 75 mg triamterene free shipping. The researchers compared the hotels on the gap between prior expectation and actual quality, using the difference score, y = performance gap = (prior expectation score - actual quality score). Identify the response variable, the factor, and the categories that form the groups. The survey asks the customers which way they use the bank the most: (1) interacting with a teller at the bank, 690 Chapter 14 Comparing Groups: Analysis of Variance Methods Customer satisfaction with outsourcing Group 2 1 5 Group 3 9 10 5 8. Explain how a lurking variable could be responsible for Group 3 having a larger mean than the others. Then the standard deviations are the same as reported in the table, but the sample means are 6, 7, and 8 rather than 6, 3, and 8. Suppose you had the same means as shown in the table but the sample standard deviations were 1. Suppose you had the same means and standard deviations as shown in the table but the sample sizes were 30, 20, and 30, instead of 3, 2, and 3. Callers rated their satisfaction on a scale of 0 to 10, with higher scores representing greater satisfaction. A recent General Social Survey asked subjects, "What is the ideal number of kids for a family? Based on part c, can you conclude that every pair of religious affiliations has different population means for ideal family size? The subjects formed three groups according to smoking status (never, former, current). Each subject completed a personality questionnaire that provided scores on various personality scales designed to have overall means of about 50 and standard deviations of about 10. The table shows some results for three traits, giving the means with standard deviations in parentheses. For the response variable, use the number of weekly hours engaged in sports and other physical exercise. Using software, for each gender construct box plots and find the mean and standard deviation of the response variable. To compare the means, suppose you instead used the two-sided t test from Section 10. Specify the value you changed, and report the resulting F test statistic and the P-value. Use these and sample summary means and standard deviations to describe the three samples. In practice, we can estimate differences between population means with confidence intervals. A ni nj the t-score from the t table has df = N - g = total sample size - # groups. Example 5 Fisher method Number of Good Friends and Happiness Picture the Scenario Chapter 11 investigated the association between happiness and several categorical variables, using data from the General Social Survey. The respondents indicated whether they were very happy, pretty happy, or not too happy. For comparing the very happy and pretty happy categories, the confidence interval for 1 - 2 is (y1 - y2) { t. Since the confidence interval contains only positive numbers, this suggests that 1 - 2 7 0; that is, 1 exceeds 2. On the average, people who are very happy have more good friends than people who are pretty happy.   Thus a and O must be the polar coordinates of the point whose cartesian coordinates are (A heart attack yawning quality triamterene 75mg, B) blood pressure chart while pregnant buy triamterene master card, and so we can always find such an a and O with a > 0 pulse pressure in shock discount triamterene line. The motion described by the solutions of the spring equation is called simple harmonic motion hypertension prevention and treatment generic triamterene 75mg line. It arises whenever a system is subject to a restoring force proportional to its displacement from equilibrium. Every solution of the spring equation - - 02x has the form x = A cos wt + B sin at, where A and B are constants. Up until now we have distinguished variables, which are mathematical objects that represent "quantities," and functions, which represent relations between quantities. Beginning with the next example, we will occasionally drop our scruples in distinguishing functions from variables and will use this abbreviated notation. Let the weight be initially extended by a distance of 1 centimeter moving at a velocity of 2 centimeters per second. We use the spring equation (3) with w = where k is the spring constant and m is the mass of M. Since k = \$ and m = 1, w is At t = 0, M is extended by a distance of 1 centimeter and moving at a velocity of 2 centimeters per second, so x, = 1 and v, = 2. The maximum displacement is the amplitude where A and B are the coefficients of sin and cos in the a solution. Such forces occur in more realistic models for springs and in equations for electric circuits. The point x, is an equilibrium position since the constant function x(t) = x, satisfies the equation of motion, by condition (i). By condition (ii), the force is positive when x is near x, and x < x, while the force is negative when x is near x, and x > x. Thus the particle is being pushed back toward x, whenever it is near that point, just as with the spring in Figure 8. Since f(xo) = 0, the equation (8) becomes which is called the linearization of equation (8) at x. We thus conclude that, to the degree that the linear approximation of the force is valid, the particle oscillates around the equilibrium point x, with It can be shown that the particle subject to the period 2n/ exact force law (8) also oscillates around x, but with a period which depends upon the amplitude of the oscillations. As the amplitude approaches zero, the period approaches 2n/ which is the period for the linearized equation. The word stable refers to the fact that motions which start near x, with small initial velocity stay near x. Using conservation of energy, one can show that the motion for the exact equations stays near xo as well. In Exercises 9-12, sketch the graph of the given function and find the period, amplitude, and phase shift. What happens to the frequency of oscillations if three equal masses are hung from a spring where there was one mass before? Find a differential equation of the "spring" type satisfied by the function y(t) = 3 cos(t/4) sin(t/4). A "flabby" spring exerts a force f(x) = -3x 2x3 when it is displaced a distance x from its equilibrium state, x = 0. An atom of mass m in a linear molecule is subjected to forces of attraction by its neighbors given by d;jl 17. If x = 0 is the equilibrium position, it is given that x = 1 and dx/dt = 1 when t = 0. Show that if the composite function f 0 g satisfies the spring equation (with the same a), then a = + 1. Exercises 31 and 32 outline the complete proof of the following theorem using the "method of variation o f constants": Let x = f (t) be a twice-differentiable function of t such that (d 2 x / d t 2)+ w2x = 0. Use the calculations in Exercise 31 to give the proof o f the theorem, making sure to avoid circular reasoning. Then rewrite formulas (1 1) and (12) as dx / dt B = x(t)sin wt + - at, cos (13) w and dx/dt A = x(t)coswt - - at. A(t)=xcoswt- + It is possible to choose A (t) and B (t) in many ways, since for each t either sinwt or cosot is nonzero. Since this is what we are trying to prove, we should be very suspicious here o f circular reasoning. Many quantities, such as bank balances, populations, the radioactivity of ores, and the temperatures of hot objects change at a rate which is proportional to the current value of the quantity.

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