2.45, 1.60, 1.53, 2.89, 2.85, 1.09, 1.23, 2.39, 2.97, 2.28, 1.79, 1.48, 1.62, 1.17, 1.41
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1 Name: Date: THE STANDARD DEVIATION ALGEBRA 2 WITH TRIGONOMETRY Measurements of central tendency, such as the mean and median, are important numbers that summarize average behavior of a data set. But, it is also important to know how much the data varies in a data set. The interquartile range from the last lesson was a rough measurement of dispersion. A more important measurement of dispersion is known as the standard deviation and can be thought of as the average distance the points of data in a set fall away from the set s mean. For the data set Eercise #1: Both of the following data sets have means of 5. Calculate the standard deviation of both data sets using the formula above and round your answers to the nearest tenth. Data Set #1: 1 2 STANDARD DEVIATION FORMULA 1 the standard deviation is given by n 2,,..., n 2, 4, 5, 9 Data Set #2: 4, 5, 5, 6 n i1 i In all practical applications, we will assume the use of the graphing calculator to find the standard deviation of a data set. In Eercise #2 we return to a data set from the last lesson. Eercise #2: Biologists are studying the weight of brown trout in upstate New York streams. They capture and weigh 15 brown trout from the Saw Kill Creek in Red Hook, New York. Their weights, accurate to the nearest hundredth of a pound, are given below: 2.45, 1.60, 1.53, 2.89, 2.85, 1.09, 1.23, 2.39, 2.97, 2.28, 1.79, 1.48, 1.62, 1.17, 1.41 (a) Determine the mean and standard deviation of the weights of the brown trout. Round both answers to the nearest hundredth of a pound. (b) Use your calculator to sort the data above in ascending order. Then, using your answers from part (a), determine the percent of brown trout weights that fall within one standard deviation of the mean. ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #3
2 Eercise #3: Which of the following data sets would have the smallest standard deviation? Do this without the use of your calculator. (1) 3, 5, 9,13,18 (3) 1, 3, 5,10,15 (2) 4, 0,1, 2, 5 (4) 4, 4, 4, 5, 5 Samples Versus Populations Many times when statistical studies are done, it is not possible to get a data value for each member of the population. In this case a sample must be taken. There is a slight difference between how the population standard deviation,, is calculated and how the sample standard deviation, s, is calculated. It is important in any problem to pay close attention to whether a sample or the entire population is being discussed. Eercise #4: A car company is trying to determine the fuel efficiency, in miles per gallon, of their latest hybrid vehicle. They drive a sample of 50 of these vehicles and determine the fuel efficiency. The data set is given in the table below. Number of (a) Using your calculator, determine the mean and standard deviation of the fuel efficiency. Round your standard deviation to the nearest hundredth. (b) Based on (a), how many cars in this sample had a fuel efficiency more than one standard deviation below the mean? Fuel Efficiency i Cars f i (c) The government will only allow the company to advertise a fuel efficiency for a vehicle that is at most two standard deviations above the mean. Can the company claim this car gets 42 miles per gallon? Eplain. Eercise #5: In which of the following would it be appropriate to use the entire population instead of a sample? (1) determining the average weight of New Yorkers (2) determining the average time spent on cell phones by American teenagers (3) determining the average quiz grade for a given Algebra 2 class (4) determining the average weight of Atlantic salmon ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #3
3 Name: Date: SKILLS THE STANDARD DEVIATION ALGEBRA 2 WITH TRIGONOMETRY - HOMEWORK 1. Which of the following data sets has the smallest standard deviation? (1) 3, 4, 4, 5 (3) 1,1, 3, 5 (2) 4, 7,10, 20 (4) 2, 5, 9,11 2. Which of the following data sets has a standard deviation of zero? (1) 2, 1, 0,1, 2 (3) 7, 7, 7, 7, 7 (2) 3, 3, 4, 5, 5 (4) 0, 0,1,1, 2 3. For the data set 3, 5, 9,12,15,18, 21, which of the following is true? (1) interquartile range 2 standard deviation (2) interquartile range 2 standard deviation (3) interquartile range 2 standard deviation (4) 1 interquartile range standard deviation 2 APPLICATIONS 4. For each of the following statistical studies, determine whether it would be more appropriate to use a sample or the entire population. (a) Determining the average cholesterol level of 40 year old Americans (b) Determining the average age of math teachers at Arlington High School (c) Determining the average weight of newborn baldheaded eagles. 5. Mr. Weiler s Advanced Calculus class took a quiz with the results shown below. Determine the mean and standard deviation of the quiz grades. Round both answers to the nearest tenth. 56, 68, 75, 77, 78, 82, 85, 86, 88, 90, 93, 98, 100 ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #3
4 6. A company is trying to determine how long teenagers speak on cell phones per day. They survey 100 teenagers and find the following distribution for the amount of time they spent on a cell phone for a given day. (a) Find the mean and standard deviation of this data set. Round both answers to the nearest tenth of a minute. (b) The company is planning on charging overage fees to any person who talks for a time greater than one standard deviation above the mean. How many of these 100 teenagers would be charged an overage fee? Show the calculations that lead to your answer. Number of Minutes i Number of Teens f i A farmer would like to determine the acidity of his soil before he plants crops. He takes 12 samples of soil from a particular field and measures their ph. The results are shown below. 5.7, 6.1, 5.8, 6.7, 7.2, 6.3, 6.8, 6.5, 5.9, 6.4, 6.0, 6.9 (a) Find the mean and the standard deviation for this data set. Round both answers to the nearest hundredth. (b) Determine, to the nearest percent, the portion of this data set that falls within one standard deviation of the mean. 8. On a recent SAT math test, the mean was 580 with a standard deviation of 60. If Jonathan scored a 655 on this particular eam, which of the following represents the number of standard deviations Jonathan s score was above the mean? (1) 1 (3) 3 (2) 1.25 (4) In a particular study, the mean heart rate for patients was 82 beats per minute with a standard deviation of 8 beats per minute. If Carlos participated in the study and his heart rate was three-quarters of a standard deviation below the mean, which of the following was his heart rate, in beats per minute? (1) 74 (3) 76 (2) 88 (4) 79 ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #3
5 Name: Date: THE NORMAL DISTRIBUTION ALGEBRA 2 WITH TRIGONOMETRY Many large data sets have a distribution that can be well described with what is known as The Normal Distribution or the Bell Curve. This curve, as seen in the accompanying handout to this lesson, shows the percent of a normally distributed data set that lies between half standard deviation increments from the mean. Eercise #1: For a population that is normally distributed, find the percentage of the population that lies (a) within one standard deviation of the mean. (b) within two standard deviations of the mean. (c) more than three standard deviations away from the mean. (d) between one and two standard deviations above the mean. As can be easily seen from Eercise #1, the majority of any normally distributed population will lie within one standard deviation of its mean and the vast majority will lie within two standard deviations. A whole variety of problems can be solved if we know that a population is normally distributed. Eercise #2: At Arlington High School, 424 juniors recently took the SAT eam. On the math portion of the eam, the mean score was 540 with a standard deviation of 80. If the scores on the eam were normally distributed, answer the following questions. (a) What percent of the math scores fell between 500 and 660? (b) How many scores fell between 500 and 660? Round your answer to the nearest whole number. (c) If Evin scored a 740 on her math eam, what percent of the students who took the eam did better than her? (d) Approimately how many students did better than Evin? ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #4
6 Eercise #3: The heights of 16 year old teenage boys are normally distributed with a mean of 66 inches and a standard deviation of 3. If Jabari is 72 inches tall, which of the following is closest to his height s percentile rank? (1) 85 th (3) 98 th (2) 67 th (4) 93 rd Eercise #4: The amount of soda in a standard can is normally distributed with a mean of 12 ounces and a standard deviation of 0.6 ounces. If 250 soda cans were pulled by a company to check volume, how many would be epected to have less than 11.1 ounces in them? (1) 17 (3) 28 (2) 23 (4) 11 Eercise #5: Biologists are studying the weights of Red King Crabs in the Alaskan waters. They sample 16 crabs and compiled their weights, in pounds, as shown below. 9.8, 10.1, 11.1, 12.4, 11.8, 13.2, 12.8, 12.5, 13.7, 11.6, 13.4, 12.3, 12.6, 14.8, (a) Determine the mean and standard deviation for this sample of crabs. Round both statistical measures to the nearest tenth of a pound. (b) Why does this sample indicate that the population would be well modeled using a normal distribution? Eplain. Hint Use your calculator to sort this data in ascending order. (c) Assuming your mean and standard deviation from part (a) apply to a normally distributed population of crabs caught in Alaska, what percent will fall between 9.6 pounds and 15.6 pounds? (d) If fishermen must throw back any crab caught below 10.4 pounds, approimately what percent of the crabs caught will need to be thrown back if the weights are normally distributed? ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #4
7 Name: Date: SKILLS THE NORMAL DISTRIBUTION ALGEBRA 2 WITH TRIGONOMETRY - HOMEWORK 1. A variable is normally distributed with a mean of 16 and a standard deviation of 6. Find the percent of the data set that: (a) is greater than 16 (b) falls between 10 and 22 (c) is greater than 28 (d) is less than 1 (e) falls between 4 and 19 (f) falls between 22 and 31 APPLICATIONS 2. The weights of Siamese cats are normally distributed with a mean of 6.4 pounds and a standard deviation of 0.8 pounds. If a breeder of Siamese cats has 128 in his care, how many can he epect to have weights between 5.2 and 7.6 pounds? (1) 106 (3) 98 (2) 49 (4) If one quart bottles of apple juice have weights that are normally distributed with a mean of 64 ounces and a standard deviation of 3 ounces, what percent of bottles would be epected to have less than 58 ounces? (1) 6.7% (3) 0.6% (2) 15.0% (4) 2.3% 4. Historically daily high temperatures in July in Red Hook, New York, are normally distributed with a mean of 84 F and a standard deviation of 4 F. How many of the 31 days of July can a person epect to have temperatures above 90 F? (1) 6 (3) 9 (2) 2 (4) 4 ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #4
8 5. The weights of four year old boys are normally distributed with a mean of 38 pounds and a standard deviation of 4 pounds. Which of the following weights could represent the 90 th percentile for the weight of a four year old? (1) 47 pounds (3) 43 pounds (2) 45 pounds (4) 41 pounds 6. The lengths of songs on the radio are normally distributed with a mean length of 210 seconds. If 38.2% of all songs have lengths between 194 and 226 seconds, then the standard deviation of this distribution is (1) 16 seconds (3) 8 seconds (2) 32 seconds (4) 64 seconds 7. The heights of professional basketball players are normally distributed with a standard deviation of 5 inches. If only 2.3% of all pro basketball players have heights above 7 foot 5 inches, then which of the following is the mean height of pro basketball players? (1) 6 feet 5 inches (3) 6 feet 10 inches (2) 6 feet 2 inches (4) 6 feet 7 inches 8. On a recent statewide math test, the raw score average was 56 points with a standard deviation of 18. If the scores were normally distributed and 24,000 students took the test, answer the following questions. (a) What percent of students scored below a 38 on the test? (b) How many students scored less than a 38? (c) If the state would like to scale the test so that a 90% would correspond to a raw score that is one and a half standard deviations above the mean, what raw score is needed for a 90%? (d) How many of the 24,000 students receive a scaled score greater than a 90%? (e) The state would like no more than 550 of the 24,000 students to fail the eam. What percent of the total does the 550 represent? Round to the nearest tenth of a percent. (f) What should the raw passing score be set at so that no more than the 550 students fail? ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #4
9 THE NORMAL DISTRIBUTION BASED ON STANDARD DEVIATION 19.1% 19.1% 15.0% 15.0% 0.1% 0.5% 1.7% 4.4% 9.2% % 4.4% 1.7% % 0.1% ALGEBRA 2 WITH TRIGONOMETRY, UNIT #12 STATISTICS LESSON #4
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