# MATH 1473: Mathematics for Critical Thinking. Study Guide and Notes for Chapter 4

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1 MATH 1473: Mathematics for Critical Thinking The University of Oklahoma, Dept. of Mathematics Study Guide and Notes for Chapter 4 Compiled by John Paul Cook For use in conjunction with the course textbook: Mathematics: A Practical Odyssey (Sixth Edition) By David Johnson and Thomas Mowry The exercises in this packet come from a variety of sources: the course textbook, and supplementary course materials provided by Ms. Christine Tinsley, University of Oklahoma; additionally, some of the exercises are original.

2 Section 4.1: Population, Sample, and Data Key Terms Statistics, descriptive statistics, inferential statistics, population, sample, data point, frequency, frequency distribution, relative frequency, histogram, density, relative frequency density, categorical data, pie chart Practice Problems 1) To study the literary habits of students at a local college, thirty randomly selected students were surveyed to determine the number of times they had been to the library during the last week. The following results were obtained: a) Organize the given data by creating a frequency distribution. b) Construct a pie chart to represent the data.

3 c) Construct a histogram using single-valued classes of data. 2) The weights, in pounds, of 35 packages of ground beef at the Cut Above Market were as follows: a) Organize the given data by creating a frequency distribution. b) Construct a histogram to represent the data.

4 3) To study the output of a machine that fills boxes with cereal, a quality control engineer weighed 150 boxes of Brand X cereal. The frequency distribution in the following table summarize her findings: x=weight (in ounces) Number of boxes 15.3 x< x< x< x< x< Construct a histogram to represent the data: 4) The frequency distribution below lists the number of hours per day a randomly selected sample of teenagers spent watching television. Where possible, determine what percent of the teenagers spent the following number of hours watching television: a) Less than 4 hours Hours per day Number of Teenagers 0 x< x< x< x< x< x< x<7 25 b) At least 1 hour

5 c) At least 2 hours but less than 4 hours 5) The following table lists some common specialties of physicians in the United States in 2002: Specialty Male Female Family practice 63,194 23,317 General surgery 32,678 4,525 Internal medicine 101,633 41,658 Obstetrics/gynecology 25,606 15,432 Pediatrics 33,020 33,351 Psychiatry 27,803 12,292 Construct a pie chart to represent the male data.

6 Writing and Understanding 6) Explain the difference between the terms population and sample. 7) Explain the difference between frequency and relative frequency. 8) When should data be grouped in intervals? What the advantages and disadvantages of this method? Section 4.1 Homework: 3, 5, 8, 9, 15, 18, 22

7 Section 4.2: Measures of Central Tendency Key terms measure of central tendency, mean, median, mode, outlier Practice Problems 1) In 2005, Lance Armstrong won his seventh consecutive Tour de France bicycle race. No one in the 100 year history of the race has won so many times. Lance s winning times are given in the following table: Year Time (minutes) Find Armstrong s mean winning time. 2) In 2001, the U.S. Bureau of Labor and Statistics tabulated a survey of workers ages and wages. The frequency distribution in Figure 4.52 summarizes the age distribution of workers who received minimum wage (\$5.15 per hour). Find the mean age of a worker receiving minimum wage. y=age Number of Workers 16 y<20 640, y<25 660, y<35 372, y<45 276, y<55 171, y<65 111,000 n=2,230,000

8 3) Ten college students were comparing their wages earned at part time jobs. Nine earned \$10.00 per hour working at jobs ranging from waiting on tables to working in a bookstore. The tenth student earned \$ per hour modeling for a major fashion magazine. Find the mean wage of the ten students. 4) Eugene must take four exams in a geography class. If his scores on the first three exams are 91, 67, and 83, what does he need on the fourth exam for his overall grade to be at least 90? 5) Find the median of the following sets of data: {2, 8, 3, 12, 6, 2, 11}, {2, 8, 3, 12, 6, 2, 11, 8},{10, 10, 10, 10, 10, 10, 10, 10, 10, 200}

9 6) The mean salary of ten employees is \$32,000, and the median salary is \$30,000. The highest paid employee gets a \$5,000 raise. a. What is the new mean salary of these ten employees? b. What is the new median salary of these ten employees? 7) Find the mode of the following sets of data: {4, 10, 1, 8, 5, 10, 5, 10}, {4, 9, 1, 10, 1, 10, 4, 9}, {9, 6, 1, 8, 3, 10, 3, 9} Writing and Understanding 8) Which of the three measures of central tendency are more susceptible to outliers? What conclusions can be drawn from this?

10 9) Suppose the mean of Group 1 is a and the mean of group 2 is b. Does this mean that the mean of the union of the two groups is (a+b)/2? Why or why not? 10) When presented with a frequency distribution with intervals (as in practice problem 2), why is it only possible to approximate the mean value? Section 4.2 Homework: 2, 5, 8, 11, 13, 14, 15, 17, 19

11 Section 4.3: Measures of Dispersion Key terms measure of dispersion, deviation, variance, standard deviation, Practice Problems 1) To settle an argument over who was the better bowler, George and Danny agreed to bowl six games, and whoever had the highest average would be considered best. Their scores are reflected in the following table: George Danny Use mean, median, and mode to determine which bowler is better. Why does this problem require a measure of dispersion, instead of just measures of central tendency? 2) Write the following formulas: a. Sample variance b. Sample standard deviation

12 3) For both George and Danny s bowling scores, find the following: a. The deviations from the mean b. The means of the deviations c. The squares of the deviations d. The sums of the squares of the deviations e. The variances f. The standard deviations 4) Using the above problem as a guide, write a step-by-step guide for finding the variance and the standard deviation of a set of data.

13 5) Write the Alternative Formula for Sample Variance: 6) Write a step-by-step guide for using the above formula: 7) Use the Alternative Formula to find the variance for both George and Danny. (Ideally, this will convince you that both formulas work!) 8) Now that we have introduced the variance and the standard deviation, which of the two bowlers is the best? Why?

14 9) Bob is a rabbit enthusiast, and he has 11 Netherland Dwarf rabbits. The weights of the rabbits (in ounces) are given in the following table: What percentage of the rabbit s weights lie within one standard deviation of the mean? Writing and Understanding 10) When studying the dispersion of a set of data, why are the deviations from the mean squared? 11) Why is it sometimes necessary to look at the dispersion in addition to the measures of central tendency? Section 4.3 Homework: 4, 5, 7, 11, 12, 14

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