of a student s grades for the period is a better method than using the mean. Suppose the table at the right shows your test grades.

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1 Math Applications The applications that follow are like the ones you will encounter in many workplaces. Use the mathematics you have learned in this chapter to solve the problems. Wherever possible, use your calculator to solve the problems that require numerical answers. 1 Some teachers suggest that using the median of a student s grades for the period is a better method than using the mean. Suppose the table at the right shows your test grades. a. Use the numerical scale of 4 for an A, 3 B 3 for a B, 2 for a C, 1 for a D, and 0 for an F. Convert your letter grades to number grades. see table above b. Determine your mean grade for the period. 2.3 c. Determine your median grade for the period. 3 Test Grades D 1 F 0 D 1 C 2 C 2 B 3 B 3 C 2 B 3 B 3 A 4 B 3 d. Based on your findings, would you rather have a mean teacher or a median teacher? median 2 The players on a professional basketball team have the following free-throw percentages at the middle of the season. Player # Free-throw % Player # Free-throw % a. Determine the median free-throw percentage for this team. 82.7% b. Determine the mean free-throw percentage for this team. 81.7% c. A competing team has a higher median free-throw percentage of 85.0% but a lower mean of 77.0%. How can this be possible? The lower half of the data are more spread out than the upper half of the data. d. Suppose your team trades Player #5 for a player with a free-throw percentage of 79.0%. How will this affect the team s median freethrow percentage? How will it affect its mean free-throw percentage? median is not changed; mean will be greater Math Applications 431

2 3 School administrators at Lincoln High School are examining the performance of their juniors and seniors on the Scholastic Aptitude Test (SAT). The data shown in the table shows the performance. Juniors Seniors Number of Students Mean Score 1,054 1,125 Median Score 1,036 1,099 Low Score High Score 1,528 1,562 Standard Deviation a. What are the ranges in scores for the juniors and seniors? Does this seem consistent with the standard deviations for the two classes of students? Explain. 697; 744; yes, the wider range would probably indicate more variation and hence a greater standard deviation. b. Assuming the SAT scores for the juniors are normally distributed, between what two scores would you expect about 95% of the juniors to fall? between 808 and 1,300 c. Assuming the SAT scores for the seniors are normally distributed, between what two scores would you expect about 95% of the seniors to fall? between 865 and 1,385 d. Consider the median test score for the senior class. Which half of the data (upper or lower) do you think is more widely dispersed? Explain. the upper half because the mean is greater than the median 4 The table contains data on the amount of money raised by a charity organization during the holiday season and the number of volunteers employed. Year Number of Volunteers Amount Raised $128, $141, $132, $134, $143, $139, $146, $142, $145, Chapter 7 Statistics

3 a. Draw a scatter plot of the data. Does there appear to be positive, negative, or no correlation between the two variables? see margin for graph; positive correlation b. Let f(v) represent the amount of money raised by v volunteers. Which linear function would best describe a line of best fit? ii i. f(v) 5 250v + 84,000 ii. f(v) 5 475v + 29,000 iii. f(v) 5 400v + 77,000 iv. f(v) 5 125v + 35,000 c. Plot the line of best fit from part b on the same graph with the data points. Does the line seem to accurately model the data? see margin for graph; yes d. If the charity organization expects to have 255 volunteers next holiday season, about how much money would you expect to be raised? Use the line of best fit to make a prediction. Round your answer to the nearest thousand dollars. about $150,000 e. How accurate do you think the line of best fit would be in predicting dollar amounts for very large or very small values of v? Is it likely to be more accurate in predicting dollar amounts for values of v between 200 and 250? Explain. It will be more accurate for values between 200 and 250. High or low values of v may lead to extrapolation errors. 5 Samples of apples from an orchard were weighed. The weights were rounded to the nearest ounce and recorded. Sample Apple Weights (oz) a. Construct a tally of the frequencies using one through ten ounces as weight classes. 4-2; 5-8; 6-6; 7-3; 8-1 b. Determine the percent frequency for each weight class. 10%; 40%; 30%; 15%; 5% c. Construct a percent frequency histogram of the data. see margin d. What are the mode, median, and mean of the apple weights sampled? 5 ounces; 5.5 ounces; 5.7 ounces Math Applications 433

4 6 The weights of grain loaded in the bins of a river barge were recorded as follows. Load Weight per Bin (ton) Barley 52.8 Corn 57.0 Flaxseed 49.8 Oats 43.4 Rice 49.1 Rye 52.6 Sorghum 56.6 Soybeans 54.8 Wheat 57.8 a. Compute the mean weight of a bin tons b. Order the data from least to greatest and make a box-and-whisker plot. 43.4, 49.1, 49.8, 52.6, 52.8, 54.8, 56.6, 57.0, 57.8; see margin for graph c. Use the plot to find the median weight of a bin tons d. Use the plot to find the range of weights for these bins tons e. Which half of the data is more widely dispersed? Do there appear to be any outliers in the data? lower half; no 7 The table below lists the average wheat yields for several countries during a certain time span. Average Wheat Yields (pounds per acre) Brazil 678 Canada 1,401 France 2,766 India 723 Japan 1,999 Mexico 2,150 Spain 1,008 USA 1,552 Russia 946 United Kingdom 3, Chapter 7 Statistics

5 a. What is the mean wheat yield? 1,683.6 pounds per acre b. Order the wheat yields from least to greatest and make a box-andwhisker plot. 678; 723; 946; 1,008; 1,401; 1,552; 1,999; 2,150; 2,766; 3,613; see margin for graph c. What is the median wheat yield? 1,477 pounds per acre d. What is the range of the wheat yields? 2,935 pounds per acre e. Which half of the data is more widely dispersed? Do there appear to be any outliers in the data? upper half; the United Kingdom wheat yield may be an outlier 8 You have asked twenty of your customers to fill out a questionnaire about the quality of your product. One question asks how satisfied they are with their purchase, on a scale of 1 (dissatisfied) to 5 (very satisfied). Customer Ratings a. Make a frequency distribution table for this data. 1-2; 2-2; 3-5; 4-8; 5-3 b. Identify the mode and the median. 4; 4 c. Compute the mean rating of these customers. Would the mean be a good way to summarize such ratings? Explain your answer. 3.4; see margin for explanation d. Which measure of central tendency helps you determine if at least half of your customers are reasonably satisfied with their purchase (that is, they give a rating of 4 or 5)? median Math Applications 435

6 9 Your retail toy business frequently experiences short-lived fads. A recent fad involved a particular type of stuffed animal. Your records reflect the rapid rise and decline of sales of this item. Week # Items Sold a. Pick a convenient interval size and draw a histogram to reflect the changing sales for each week. see margin b. Find the mode for the frequency distribution. week 4 c. Why would the mode of a distribution like this be of interest to a store owner? to determine when sales will reach their peak d. Suppose the mode occurred later for the next fad. What might the store owner assume about the sales for the weeks that will follow? Is this a valid assumption? see margin 10 You have recorded the end-of-the-week inventories for your clothing boutique over the past 28 weeks. End-of-the-Week Inventory a. Make a stem-and-leaf plot of the end-of-the-week inventory data. see margin b. Determine the mode, median, and mean of this data. 22; 22; about 23.4 c. Do there appear to be any outliers in the inventory values in the stemand-leaf plot? If so list the outlier(s) yes; 46 d. Your goal for the end-of-the-week inventory is at least 19 items. How is the median the most useful measure of central tendency in evaluating this goal? You have met your goal in at least half the weeks. 436 Chapter 7 Statistics

7 11 The drip rate of a patient s IV changes depending on the position of the patient s arm. Over the course of a couple of hours, you count and record the number of drops per minute several times. Number of Drops per Minute a. Make a frequency distribution of the drip rates. 8-1; 9-1; 10-3; 11-4; 12-1 b. What are the values of the mean, mode, and median of this data? 10.3 drops per minute; 11 drops per minute; 10.5 drops per minute c. What is the range of these data? 4 drops per minute 12 A consumer magazine performed a study on the leading brands of black tea, analyzing the caffeine content of a cup of each brand. The table lists the findings, in milligrams of caffeine per cup of brewed tea. Brand Caffeine (mg/cup) Brand Caffeine (mg/cup) a. Group the data by caffeine content into classes of 10 to 19, 20 to 29, and so on, and draw a histogram of the data. see margin b. Identify the mode and median of this data. 74 milligrams per cup; 62 milligrams per cup c. Determine the mean of the data and compare it to the mode and median you determined above. about 58.5 milligrams per cup; does not closely agree Math Applications 437

8 13 You are surveying the prices for carpet padding for a client. You record the following prices from several different suppliers. Carpet Padding Price ($ per sq. yd) $2.09 $3.55 $2.75 $3.15 $2.99 $3.07 $2.95 a. What is the range in carpet padding prices? $1.46 b. What is the mean carpet padding price? $2.94 c. What is the median carpet padding price? $ You have recorded the temperatures from a refrigerated meat case during a six-day period, twice each day. Temperature ( F) Day Day Day Day Day Day a. What is the range of the temperatures? 2 F b. What is the mean temperature? about 38.9 F c. What is the median temperature? 39 F d. What is the mode of the temperature data? 38 F 438 Chapter 7 Statistics

9 15 A manufacturing company produces circuit boards for electronic devices such as alarm clocks, calculators, and radios. The quality control department wants to reduce the number of defective boards produced. Specifically, they are interested in determining whether there is a relationship between the number of defective boards produced by new employees and the number of training hours the employee received after being hired. Employee Number Hours of Training Defective Boards a. The table shows the number of defective circuit boards produced by several new employees during their first month along with the number of training hours the employees received. Use your calculator to make a scatterplot of the data. see margin b. Is there a correlation between the two variables in the data? If so, is it a positive or a negative correlation? yes; negative c. What conclusion might the quality control department draw regarding the number of training hours received by new employees and the number of defective circuit boards they produce? The more training a new employee receives, the fewer defective boards will be produced. d. If a line of best fit were drawn to model the data, would it likely have a positive or a negative slope? negative Math Applications 439