unit II q12. Summarized on the right is the sale (in thousands) of the top five products in 2002. Use this information to calculate what percent of the sales of these five products was due to product D. of the sales of these five products was due to product D.(Round to the nearest hundredth as needed.) Qtn 21. Use the formula for computing future value using compound interest to determine the value of an account at the end of 10 years if a principal amount of $2,500 is deposited in an account at an annual interest rate of 5% and the interest is compounded daily. (Assume there are 365 days in a year.) The amount after 10 years will be $ (Round to the nearest cent as needed.) Unit II 28. Angela's bank gave her a 2year addon interest loan for $5,090 to pay for new equipment for her antiques restoration business. The annual interest rate is 10.44%. How much interest will she pay? What are her monthly payments? She will pay $ in interest on the loan. (Round to the nearest cent.) Unit II 29. Using the unpaid balance method, find the current month's finance charge on a credit card account having the following transactions. Last month's balance: $625 Last payment $120 Annual Interest rate: 18% Purchases 399 Returns: $414 The finance charge is $ (Round to the nearest cent.) Unit II 30. Use the average daily balance method to compute the finance charge on the credit card account for the month of August (31 days). The starting balance from the previous month is $280. The transactions on the account for the month are given in the table to the right. Assume an annual interest rate of 22% on the account and that the billing date is August 1st. Date Transaction August 7 Made payment of $71 August 13 Charged $140 for hiking boots August 22 Charged $27 for gasoline August 26 Charged $25 for restaurant meal The finance charge for the month of August is $ . (Round to the nearest cent as needed.) Unit II 31. Find the APR of the loan given the amount of the loan, the number and type of payments, and the addon interest rate. Loan amount, $6,000; three yearly payments; rate=8% The annual percentage rate is (Type an integer or a decimal.) Unit 32. Find the finance charge per $100 for the loan described below. Loan, $2,400; finance charge, $384. The finance charge is $ per $100. Unit II 33. Luisa pays a finance charge of $152 on a 6month, $3,700 loan. Find the annual percentage rate using the annual percentage rate table APR 10% 11% 12% 13% 14% 15% 16% Number of Payments Finance Charge per $100 $2.94 $3.23 $3.53 $3.83 $4.12 $4.42 $4.72 6 12 $5.5 $6.06 $6.62 $7.18 $7.74 $8.31 $8.88 0 24 $10.7 $11.86 $12.98 $14.10 $15.23 $16.37 $17.51 5 36 $16.1 $17.86 $19.57 $21.30 $23.04 $24.80 $26.57 6 48 Annual Percentage Rate Table $21.7 $24.06 $26.40 $28.77 $31.17 $33.59 $36.03 4 rate. Use the formula APR 2nr Over n plus 1 n=42; r=7% APR (Round to two decimal places as needed.) Unit III 11. Simplify by rationalizing the denominator. 2 5 The annual percentage rate is (Round to the nearest integer as needed.) Unit II 34. Estimate the annual percentage rate for the addon loan using the given number of payments and annual interest The answer is (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) Unit III 17. The equation D = 2H approximates the distance D, in miles, that a person can see to the horizon at a height of H feet. How many miles could a forest ranger see to the horizon if his eyes are 6 feet above footlevel and he is standing on a fire tower that is 294 feet high? Express the answer as a radical expression and then as a decimal. First express the answer as a radical expression. The forest ranger could miles to the horizon (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) Unit III 22 2 . Multiply.6 6 3 = 6 Unit III 28. Convert to decimal notation. 9.3410 = 2 4 unit III 30. Multiply and write the result in scientific notation. (2 10 )(310 ) 2 4 (210 )(310 )= (Simplify your answer. Use scientific notation. Use the multiplication symbol in the math palette as needed.) Unit iii 34. Identify the sequence as either arithmetic or geometric. List the next two terms of the sequence. 2, 1/4, 1/32, 1/256, ... Is the sequence arithmetic or geometric? Unit iii 35. For the following arithmetic sequence: a) find the specified term an and b) find the sum of the terms from a1 to an, inclusive. 4, 6, 8, 10,...; find a10. find a10.= Unit iii 40. Given the two terms in the Fibonacci sequence shown below, find the specified term. F37=24,157,817, F38=39,088,169, find F36 UNIT VII STUDY GUIDE Descriptive Statistics Course Learning Outcomes for Unit VII Upon completion of this unit, students should be able to: 1. Apply mathematical applications used in real-world situations. 1.1 Apply measures of central tendency to compare data. 6. Calculate basic statistical measures and analyze distribution graphs. 6.1 Identify the difference between a sample and a population. 6.2 Represent data visually using a variety of methods including frequency tables and stem and leaf displays to compare data. 6.3 Compute the range of a data set. 6.4 Identify how the standard deviation measures the spread of a distribution. 6.5 Apply the coefficient of a variation to compare the standard deviations of different distributions. Reading Assignment Chapter 14: Descriptive Statistics: What a Data Set Tells Us Section 14.1: Organizing and Visualizing Data, pp. 702-713 Section 14.2: Measures of Central Tendency, pp. 714-726 Section 14.3: Measures of Dispersion, pp. 727-737 Unit Lesson Suppose you owned a coffee shop in a nearby neighborhood. In order to boost sales, it would be important to know the time of day most people drink coffee or what their most popular coffee flavor is. To find these answers, we must use the principles of descriptive statistics. Descriptive statistics use data to calculate different facts about certain populations. Not all statistics are meaningful when taken out of context, but some statistics can give us an idea of surroundings, people, events, or other areas of interest. 14.1 Organizing and Visualizing Data: In this section, we will introduce the study of statistics and some key terms that are used when describing statistics. Populations and Samples Statistics is the study of gathering, organizing, analyzing, and making predictions from numerical information called data. For example, assume that we wanted to find out how many men in Florida drink coffee. First, we would need to gather the information. One way to gather information would be to conduct a survey to all men in Florida asking them if they drink coffee. Next, we would need to organize the data by identifying how many men answered yes to drinking coffee. Then, we would analyze the data by finding the percentage of men that drink coffee. Finally, we could make a prediction. For instance, if we found that 65% of men drink coffee in Florida, we could make a prediction that says that the majority of all men in the United States drink coffee. MAT 1301, Liberal Arts Math 1 We will use the study below to introduce some key terms. These key terms are important to understand UNIT x STUDY GUIDE when performing any statistical study. Title Study: Suppose that several colleges in the southeast are considering building dog parks on campus. The college presidents want to know if a dog park would be utilized. First, we are tasked to find the percentage of college students that own dogs. Key Term Definition Example Survey A tool that is administered to a sample population and used to find data about the study. A tool used to ask college students if they own a dog. Population The entire group of people mentioned in the study. All college students who attend college in the southeast Sample A part or subset of population that will be involved in the study. College students who attend the Florida State University and the University of Central Florida. *Samples are used because it is not practical to survey every person in the population. *Samples should be chosen carefully because the sample represents the entire population. Frequency Tables After a study is conducted, we need to gather the data. Usually, the data consists of large sets of numerical information. In this section, we will learn how to use frequency tables to gather and organize the data in a meaningful way. There are two types of frequency tables: frequency table and relative frequency table. A frequency table shows how many times a certain value occurs. A relative frequency table shows the percent of the time that each item occurs. The figure below represents an example of a frequency table. It shows how many students in a class received a particular grade letter. Notice that the column on the right identifies the different types of grade letters that received and the column on the left identifies how many students received that letter. Grade Letter A B C D F Total Frequency 18 24 15 12 6 75 The next figure represents an example of a relative frequency table. Relative frequency is found dividing the frequency of a value by the total of all frequencies. We will use the same example represented by the figure above to find the relative frequencies of each grade letter. MAT 1301, Liberal Arts Math 2 Grade Letter A Frequency 18 B 24 C 15 D 12 F 6 Total 75 UNIT x STUDY GUIDE Relative Frequency Title 18 = 0.24 75 24 = 0.32 75 15 = 0.2 75 12 = 0.16 75 6 = 0.08 75 1.00 Example: Construct a frequency table and a relative frequency table using the data given below: 7,8,6,5,7,10,2,7,9,5,8,8,10,9,6,5,10,7,9,8. Solution: First, we will need to construct a frequency table. To do this, draw a table and list all the different values you see in the right column. We will label the column 'x' and list the given values in order from smallest to largest. x Frequency 2 3 4 5 6 7 8 9 10 Note: The 'x' values should also be listed in consecutive order to not confuse the reader. Start with 2 in the table because it is the smallest in the set. Stop at 10 because it is the largest value in the set. Include 3 and 4 because the values are in increments of one. MAT 1301, Liberal Arts Math 3 Next, list the number of times each value occurs in the frequency column to finish frequency table: UNITthe x STUDY GUIDE Title x Frequency 2 3 4 5 6 7 8 9 10 1 0 0 3 2 4 4 3 3 The problem also asks to construct a relative frequency table. To do this, use the frequency we just found. We will add an additional column and row: x Frequency 2 1 3 0 4 0 5 3 6 2 7 4 8 4 9 3 10 3 Total 20 MAT 1301, Liberal Arts Math Relative Frequency 1 = 0.05 20 0 =0 20 0 =0 20 3 = 0.15 20 2 = 0.10 20 4 = 0.20 20 4 = 0.20 20 3 = 0.15 20 3 = 0.15 20 1.00 4 Our answer includes only two columns: 'x' and relative frequency. Therefore, our formal answer is written as UNIT x STUDY GUIDE follows: Title x Relative Frequency 1 = 0.05 20 0 =0 20 0 =0 20 3 = 0.15 20 2 = 0.10 20 4 = 0.20 20 4 = 0.20 20 3 = 0.15 20 3 = 0.15 20 2 3 4 5 6 7 8 9 10 Sometimes, we will gather data that contains many different values. If this happens, it may be beneficial to construct a frequency table based on grouping. Example: The individual race times (in minutes) are provided for 20 people. Construct a frequency table to represent the data. 12, 48, 34, 33, 20, 18, 15, 42, 23, 20, 19, 30, 32, 33, 48, 15, 20, 23, 39, 24 Solution: The smallest value in our list is 12. The largest value in our list is 48. If we listed the values in consecutive order by a unit one (as in the previous example), we would have 48 - 12 = 36 rows in our frequency table. Therefore, we will group the values and list the grouping in the right column of frequency table. You may group the data by any increment; however, the larger the increment, the less rows we will have to place in our frequency table. Our smallest value is 12, so we can round down to 10 because 10 is divisible by 5. Therefore, we can group by increments of 5. We will first start with our smallest value of 12 and round down to 10. Ten will be our starting value to our first group. We want to include five numbers in each group. Start at 10 and count until we have five numbers: 10, 11, 12, 13, and 14. This tells us that our ending value in group one is 14. Therefore, group one is 10-14. Place this group in your frequency table and count the number of times that 10, 11, 12, 13, or 14 appear in the data set. Race Times (minutes) 10-14 MAT 1301, Liberal Arts Math Frequency 1 5 Next, find the second group. To do this, list the number that comes after 14. The number that GUIDE comes after 14 UNIT x STUDY is 15. Therefore, the starting number in group two is 15. Now, count until five numbers Title are listed: 15, 16, 17, 18, and 19. Our ending number is 19. Therefore, group two is listed as 15-19. Place this in the frequency table and count the number of times 15, 16, 17, 18, or 19 appears. Race Times (minutes) 10-14 15-19 Frequency 1 4 Continue this pattern until all values are included in the table. The final solution is listed below. Race Times (minutes) 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 Frequency 1 4 6 0 5 1 1 2 Representing Data Visually Bar graphs and histograms are charts that allow the viewer to get a snapshot of the data that was collected from a particular study. Graphs also allow the viewer to make some observations about the data in a quick and easy way. Example: Construct a bar graph for the data given in the frequency table below that contains the weight loss by 30 participants in the Biggest Loser show. Weight Loss (lbs.) Frequency 0 2 1 3 2 3 3 0 4 1 5 6 6 1 7 4 8 3 9 2 10 5 Solution: To construct a bar graph, list the frequencies on the vertical axis and the pounds lost on the horizontal axis. Graphs always start at zero and count up. Label the vertical axis (frequency) with numbers 0 through 6 since 6 is the largest frequency. Label the horizontal axis (pounds lost) with numbers 0 through 10. MAT 1301, Liberal Arts Math 6 Next, represent the frequency of each pound lost by drawing a bar. The final solution is shownGUIDE below: UNIT x STUDY Title As stated before, visualizing data in the form of a graph makes it easy to make observations. For example, we can see that the most amount of people (6 people) lost 5 pounds. We can also easily see that no one lost 3 pounds. Example: Because of budget cutbacks, the campus rec center surveyed the number of students using the center hourly between 9 PM and midnight for a semester to decide if it should reduce its hours. Use the results in the given graph to answer the following questions. a. What was the smallest number of students in the rec center and how many times did it occur? 0 occurred seven times b. What was the smallest student count between 5 and 8 inclusive? How many times did that occur? The smallest student count between 5 and 8 is 7. This only occurred once between 5 and 8. c. For how many hours was the survey taken? We will find the total of all frequencies to solve: 7+8+5+3+8+10+7+9+5+3+2=67 d. For what fractional part of the total number of hours were there less than five students present? + + + = MAT 1301, Liberal Arts Math 7 Stem and Leaf Plots UNIT x STUDY GUIDE Title Stems and leaf plots are another way to organize data. These plots separate the number into two parts. We call these parts the stem and the leaf. For example, the number 89 has two parts: 8 and 9. We call the 8 a stem and the 9 a leaf. As shown, the parts are separated by the place value of each digit that make up the number. Example: Construct a stem leaf plot for the following values: 89, 87, 82, 83. Solution: First, identify the stems. The stem of a number is the first digit on the left. Therefore, every number in the set has a stem of 8. Next, identify the leaves. The leaves are the second digit of the number. Therefore, the leaves of the set are 9, 7, 2 and 3. We will write the stem on the left side of the diagram and the leaves on the right side from smallest to largest. Our answer is: Example: Represent the two sets of data on a single stem-and-leaf display. A: 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33, 42, 18, 22, 39, 21, 26,18, 43 B: 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21, 29, 22, 24, 33, 47, 23, 22, 18, 33 Solution: The stem of each number listed is the first digit. The leaves are the last digit. Since we are constructing a plot for two sets of data, we need to form the plot so that all the stems are in the middle of the plot. The leaves for set A will be on the right side of the stem, and the leaves of set B will be on the left side of the stem. First, list all the stem values. The stem values are the first digit of every number. Do not repeat the values in the chart. These values are highlighted in red. A: 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33, 42, 18, 22, 39, 21, 26, 18, 43 B: 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21, 29, 22, 24, 33, 47, 23, 22,18, 33 The unique stem values are 1, 2, 3, and 4. Second, list all the leaves in set A and B. Make sure to pair them with their corresponding stem, and list the leaves with the smallest digit closest to the stem. Our answer is: MAT 1301, Liberal Arts Math 8 14.2 Measures of Central Tendency: Now that we know how to organize data, wexcan start calculating UNIT STUDY GUIDE different statistics. The calculations we will discuss in this section are mean, median, Title and mode. The mean, median, and mode are called measures of central tendency are used by statisticians to describe different sets of data. Mean The mean of a set of data is the average. The mean of set of numbers is found by dividing the sum of all numbers by the number of values in the set. Example: Find the mean of the following data set: 3, 13, 2, 6, 9, 10, 16, 8 Solution: First, find the sum of all the numbers. 3 + 13 + 2 + 6 + 9 + 10 + 16 + 8 = 67 Next, find the number of values in the set. There are 8 numbers in the set. Last, divide 67 by 8 to get the mean. 67 8 = . Example: Exam scores: Assume that in your history of film class you have earned test scores of 78, 82, 56, and 72, and only one test remains. If you need a mean score of 70 to earn a C, then what must you obtain on the final test? Solution: Let be the grade on the final test. We then construct an equation using the formula for finding an average. 78+82+56+72+ 5 288+ 5 = 70 = 70 Plug in the 4 known test values, for the missing test, 5 because there are five test scores to be averaged, and 70 for the average. Add the numbers in the top of the fraction. 288 + = 70 5 Multiply each side of the equation by 5 to clear the fraction. 288 + = 350 Simplify the right side by multiplying. = 350 288 Subtract 288 from both sides to isolate . = 62 Simplify by subtracting on the right side to find the value of . You must make a 62 or higher on the final test to earn a C. MAT 1301, Liberal Arts Math 9 Notation UNIT x STUDY GUIDE Title Mathematicians developed a formula to express the mean. This formula uses a Greek symbol ( ) to show that we need to find the sum of a set of numbers. This symbol, , is capital sigma and is called the summation symbol. Example: Let x = 3, 7, 8, 9. Find . Solution: To find , we will add all values together in set x. Therefore, The formula for the mean is: Median The median is the middle number of a data set. It is found by listing the values of a set in order from smallest to largest and then finding the middle value. There are two cases to consider: 1. If there is an odd number of values, then the median is the value in the middle position. 2. If there is an even number of values, then the median is in the average of the two middle values. Example: Let x = 3, 7, 8, 2, 1. Find the median. Solution: List all values from smallest to largest. 1, 2, 3, 7, 8 MAT 1301, Liberal Arts Math 10 The middle value is the median, since there is an odd number of values in theUNIT data set. x STUDY GUIDE Title Example: Let x = 3, 7, 8, 9. Find the median. Solution: List all values from smallest to largest. 3, 7, 8, 9 The median is the average of the two middle values because there is an even number of values in the data set. The middle values are 7 and 8. Find the average by adding the values together and dividing by two. = 7 + 8 15 = = . 2 2 The median of the data set is 7.5. Mode The mode is the number that occurs most often in a data set. If two numbers occur the most often, then the data set has multiple modes. If the more than two values occur most often, then there is no mode. Example: Let x = 3, 2, 5, 8, 12, 3, 5, 12, 12, 2, 2 Solution: There are two modes, namely 2 and 12. They both occur 3 times. Example: Find the mean, median, and mode for the following: 12, 4, 4, 8, 4, 7, 9, 8, 7, 7 Solution: First place the data in order: 4, 4, 4, 7, 7, 7, 8, 8, 9, 12 Calculate the mean: = MAT 1301, Liberal Arts Math 4 + 4 + 4 + 7 + 7 + 7 + 8 + 8 + 9 + 12 70 = = =7 10 10 11 Calculate the median: Since the data set contains an even number of elements, wexaverage two numbers UNIT STUDY the GUIDE in the middle to find the median. Title 7 + 7 14 = =7 2 2 Calculate the mode: The highest occurring numbers (4 and 7) each occur twice. The modes are 4 and 7. Using Frequency Tables to Compute Mean, Median, and Mode It can be time consuming to calculate the mean and median of large sets of data. If the data can be organized in a frequency table, it is much faster to find these values. Steps for computing the mean of a frequency distribution - The mathematical formula for finding the mean of a frequency distribution is = ( ) The steps will demonstrate how to find the mean of the following frequency table. x 2 3 4 5 Frequency 18 32 4 12 1. Create a new column in the frequency table and label it \"Product.\" Multiply the number in the \"x\" column by the number in the frequency column. Do this for all rows in the frequency table. x 2 3 4 5 Frequency () 18 32 4 12 Product ( ) 2 18 = 36 3 32 = 96 4 4 = 16 5 12 = 60 2. Create a new row in the frequency table and label it \"Total.\" Add all of the products together and place this number in the table. This is notated by ( ). x 2 3 4 5 Total Frequency () 18 32 4 12 Product ( ) 2 18 = 36 3 32 = 96 4 4 = 16 5 12 = 60 ( ) = 36 + 96 + 16 + 60 = MAT 1301, Liberal Arts Math 12 3. Find the sum of all the frequencies in the table. UNIT x STUDY GUIDE Title x 2 3 4 5 Total Frequency () 18 32 4 12 = 66 Product ( ) 2 18 = 36 3 32 = 96 4 4 = 16 5 12 = 60 36 + 96 + 16 + 60 = 208 4. Divide the number you found in step two (208) by the number you found in step three (66). = ( ) = 208 66 = . Steps for computing the median using a frequency table - The steps will demonstrate how to find the median of the following frequency table. x 2 3 4 5 Frequency 18 32 4 12 1. Create a row and label it \"Total.\" Add all frequencies together and place the sum in the table. x 2 3 4 5 Total Frequency 18 32 4 12 66 2. Divide the total by 2. 66 2 = 3. Step 2 tells us that are 33 values before and after the median. Since the value in step two is even (66), we need to find the number in the data set that is in position 33 and 34. x 2 3 Frequency 18 32 There are eighteen 2s and thirty-two 3s. Therefore, position 33 is a 3 and position 34 is a 3. = MAT 1301, Liberal Arts Math 3+3 6 = = 2 2 13 Computing the mode using a frequency table - The mode is the 'x' value that has the highest frequency. UNIT x STUDY GUIDE Title x 2 3 4 5 Frequency 18 32 4 12 The mode is 3. Example: Find the mean, median, and mode for the following frequency distributions. Solution: Calculate the mean: The mean is found using the following formula: = ( ) = Take the value from the left column and multiply by the corresponding value in the right column. ( ) = 2 3 + 5 2 + 7 3 + 8 2 + 9 1 + 10 5 + 11 2 = 6 + 10 + 21 + 16 + 9 + 50 + 22 = 134 To find , add up all the values from the right column (column ) = = 3 + 2 + 3 + 2 + 1 + 5 + 2 = 18 ( ) 134 = = . 18 Calculate the median: We need to find the middle two numbers from the table. Since the table has 18 numbers, then the median is the average of the 9th and 10th numbers. By adding up the frequency values from the right column of the table we find that the 9th and 10th numbers are both 8. Therefore, the median is the average of 8 and 8. 8 + 8 16 = = 2 2 Calculate the mode: The number we the highest frequency is 10. MAT 1301, Liberal Arts Math 14 The Five Number Summary UNIT x STUDY GUIDE Title The Five Number Summary consists of five numbers that describe a data set. More specifically, it identifies the minimum value, first quartile (written Q1), median, third quartile (written Q3), and maximum value of a data set. The five-number summary is represented by a graph called the box-and-whisker plot. An example is shown below. Graph depicting ages (Pirnot, 2014, p. 721) Example: Construct a box and whisker plot using the following data: 6, 20, 9, 18, 17, 25, 20, 4, 22, 25, 13, 5, 6, 9, 13, 23, 20, 13, 18, 14 Solution: First, arrange the data in ascending order. 4, 5, 6, 6, 9, 9, 13, 13, 13, 14, 17, 18, 18, 20, 20, 20, 22, 23, 25, 25 MAT 1301, Liberal Arts Math 15 Next, take the data set and find the middle number or numbers and then find the middle of theGUIDE lower half (Q1) UNIT x STUDY and the middle of the upper half (Q3). Title Because our data set has an even number of elements, the median will be the average of the two numbers in the middle: = 14 + 17 = 15.5 2 We now have the information for the five-number summary and can use this information to construct a box plot. Minimum value: 4 Q1: 9 Median: 15.5 Q3: 20 Maximum value: 25 14.3 Measures of Dispersion: In this section, we will review how to find the range, standard deviation, and coefficient of variation of a dataset. These are statistics used when describing the dispersion or spread of data. The Range of Data Set The range of data set is found by subtracting the smallest value from the largest value of the data set. Example: Find the range for the following data set. {5, 7, 9, 4, 6, 8, 7, 10} Solution: First, rearrange the data and place it in order from smallest to largest. {4, 5, 6, 7, 7, 8, 9,10} MAT 1301, Liberal Arts Math 16 Next, calculate the range by subtracting the smallest value from the largest value asxfollows: UNIT STUDY GUIDE - = 10 - 4 = Title The range is 6. Standard Deviation The standard deviation also measures how much distance is between each data value and the mean of the data set. The formula for the standard deviation is as follows: Example: Find the standard deviation for the following data set. {5, 7, 9, 4, 6, 8, 7, 10} Solution: We will use the following formula for the finding the standard deviation: ( )2 = 1 Steps: 1. Calculate the mean ( ). First, find the sum of all values in the set. = 4 + 5 + 6 + 7 + 7 + 8 + 9 + 10 = 56 Next, divide the sum by the number of values in the set (n = 8). Therefore, = MAT 1301, Liberal Arts Math 17 2. Create a table. List the data values in one column and label the next two columns as follows: UNIT x STUDY GUIDE Title 3. Complete column two ( ). To do this, subtract the mean ( = 7) from every data value in column one. It is ok if you get negative values. 4. Complete column three: ( )2 . To do this, square the value you calculated in column two. *Note: To square a value, multiply that number to itself. MAT 1301, Liberal Arts Math 18 5. Plug in the values you calculated in the standard deviation formula and solve. UNIT x STUDY GUIDE Title The standard deviation is 2. We can also use what we have learned about frequency tables to calculate the standard deviation. Example: The following frequency table summarizes the number of job offers made to graduates of a Microsoft network administrator certification program. Complete the table entries to find the mean and standard deviation for this distribution. Solution: 1. Calculate the mean: 2. Complete the table as follows: To find the standard deviation we will use the following formula. Remember that the number of data values that are represented by is . The standard deviation is 1.64. MAT 1301, Liberal Arts Math 19 Example: UNIT x STUDY GUIDE Title The following table gives the annual incomes for eight families, in thousands of dollars. Find the number of standard deviations family H's income is from the mean. Solution: We will let x represent the annual income. First calculate the mean of the data set. Recall that the mean is found by adding up all values in the set and dividing by the number of values. Next, create and complete a table as shown below: Last, calculate the standard deviation by using the following formula and substitute our values: To find the number of standard deviations family H's income is from the mean, we will subtract the mean from family H's income, then divide that result by the standard deviation as follows: 51 49 = 1.25 1.60 Therefore, family H's income is 1.25 standard deviations above the mean. MAT 1301, Liberal Arts Math 20 The Coefficient of Variation UNIT x STUDY GUIDE Title The coefficient of variation is used when comparing the standard deviation of two data sets. The data values of a set are more varied, if the coefficient of variation is large. The formula for the coefficient of variation is = 100%. Therefore, to calculate the coefficient of variation (CV), divide the standard deviation by the mean and multiply by 100. The CV will always be a percentage. Example: A particular brand of laptop was sampled with regard to the time it can be used before it requires recharging. The mean time was calculated to be 7.8 hours with a standard deviation of 1.78 hours. Calculate the coefficient of variation for this example. Solution: Use the formula for the coefficient of variation: = 100% Now, substitute the given values into the equation to find the answer as follows: = 1.78 100% = 100% = 22.82% 7.8 Reference Pirnot, T. L. (2014). Mathematics all around (5th ed.). Boston, MA: Pearson. MAT 1301, Liberal Arts Math 21 unit II q12. Summarized on the right is the sale (in thousands) of the top five products in 2002. Use this information to calculate what percent of the sales of these five products was due to product D. of the sales of these five products was due to product D.(Round to the nearest hundredth as needed.) Qtn 21. Use the formula for computing future value using compound interest to determine the value of an account at the end of 10 years if a principal amount of $2,500 is deposited in an account at an annual interest rate of 5% and the interest is compounded daily. (Assume there are 365 days in a year.) The amount after 10 years will be $ (Round to the nearest cent as needed.) 4121.66 Unit II 28. Angela's bank gave her a 2year addon interest loan for $5,090 to pay for new equipment for her antiques restoration business. The annual interest rate is 10.44%. How much interest will she pay? What are her monthly payments? Monthly payments will be $256.37 She will pay $ in interest on the loan. (Round to the nearest cent.) 1062.79 Unit II 29. Using the unpaid balance method, find the current month's finance charge on a credit card account having the following transactions. Last month's balance: $625 Last payment $120 Annual Interest rate: 18% Purchases 399 Returns: $414 The finance charge is $ (Round to the nearest cent.) 7.35 Unit II 30. Use the average daily balance method to compute the finance charge on the credit card account for the month of August (31 days). The starting balance from the previous month is $280. The transactions on the account for the month are given in the table to the right. Assume an annual interest rate of 22% on the account and that the billing date is August 1st. Date Transaction August 7 Made payment of $71 August 13 Charged $140 for hiking boots August 22 Charged $27 for gasoline August 26 Charged $25 for restaurant meal The finance charge for the month of August is $ .(Round to the nearest cent as needed.) 4.5 Unit II 31.Find the APR of the loan given the amount of the loan, the number and type of payments, and the addon interest rate. Loan amount, $6,000; three yearly payments; rate=8% The annual percentage rate is (Type an integer or a decimal.) 8% Unit 32.Find the finance charge per $100 for the loan described below. Loan, $2,400; finance charge, $384. The finance charge is $ per $100. 16 Unit II 33. Luisa pays a finance charge of $152 on a 6month, $3,700 loan. Find the annual percentage rate using the annual percentage rate table APR 10% 11% 12% 13% 14% 15% 16% Number of Payments Finance Charge per $100 Annual Percentage Rate Table 6 $2.94 $3.23 $3.53 $3.83 $4.12 $4.42 $4.72 12 $5.50 $6.06 $6.62 $7.18 $7.74 $8.31 $8.88 24 $10.7 $11.86 $12.98 $14.1 $15.23 $16.3 $17.51 5 0 7 36 $16.1 $17.86 $19.57 $21.3 $23.04 $24.8 $26.57 6 0 0 nearest integer as needed.) 48 $21.7 $24.06 $26.40 $28.7 $31.17 $33.5 $36.03 4 7 9 Unit II 34.Estimate the annual percentage rate for the addon loan using the given number of The annual percentage rate is (Round to the 14% payments and annual interest rate. Use the formula APR 2nr Over n plus 1 n=42; r=7% APR (Round to two decimal places as needed.) 5.88 Unit III 11.Simplify by rationalizing the denominator. 2 5 The answer is (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) (2 5)/5 Unit III 17. The equation D = 2H approximates the distance D, in miles, that a person can see to the horizon at a height of H feet. How many miles could a forest ranger see to the horizon if his eyes are 6 feet above footlevel and he is standing on a fire tower that is 294 feet high? Express the answer as a radical expression and then as a decimal. First express the answer as a radical expression.The forest ranger could miles to the horizon 106 OR 24.49miles (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) Unit III 22 2 . Multiply.6 6 3 = 65=7776 6 Unit III 28.Convert to decimal notation.9.3410 = 9340000 2 4 unit III 30. Multiply and write the result in scientific notation.(2 10 )(310 ) 2 4 (210 )(310 )= (Simplify your answer. Use scientific notation. Use the multiplication 6.0*10^6 symbol in the math palette as needed.) Unit iii 34.Identify the sequence as either arithmetic or geometric. List the next two terms of the sequence. 2, 1/4, 1/32, 1/256, ... Is the sequence arithmetic or geometric? Geometric because the sequence has a common ration of 1/8 Unit iii 35. For the following arithmetic sequence: a) find the specified term an and b) find the sum of the terms from a1 to an, inclusive.4, 6, 8, 10,...; find a10.find a10.= an=a1+(n1)d =4+(101)2=22, d is the common difference in the sequence Unit iii 40.Given the two terms in the Fibonacci sequence shownbelow, find the specified term. F37=24,157,817, F38=39,088,169, find F36 F36=39,088,16924,157,817=14930352