1 Answer Key Name: Date: THE NORMAL DISTRIBUTION COMMON CORE ALGEBRA II Many populations 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 or proportion of a normally distributed data set that lies certain amounts from the mean. Exercise #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 Exercise #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. Exercise #2: At Arlington High School, 424 juniors recently took the SAT exam. On the math portion of the exam, the mean score was 540 with a standard deviation of 80. If the scores on the exam were normally distributed, answer the following questions. (a) What percent of the math scores fell between 500 and 660? For 500: 0.5 's below the mean For 660: 1.5 's above the mean 80 Thus we have % (b) How many scores fell between 500 and 660? Round your answer to the nearest whole number. We now need to find 62.4% of 424: students (c) If Evin scored a 740 on her math exam, what percent of the students who took the exam did better than her? For 740: 2.5 's above the mean 80 Only % scored higher. (d) Approximately how many students did better than Evin? To find 0.6% of 424 we calculate: students did better
The Normal Distribution Common Core Algebra 2 Homework Answers
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4 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? According to the normal distribution, the 90 th (1) 47 pounds (3) 43 pounds percentile must lie between 1 and 1.5 standard deviations above the mean: (3) (2) 45 pounds (4) 41 pounds x1 42 and x 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 38.2% represents the amount of a data set that must fall within 0.5 standard deviations of the mean. Thus represents one-half of a standard deviation. (2) 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? 2.3% represents the amount of a data set greater than 2 standard deviations above the (1) 6 feet 5 inches (3) 6 feet 10 inches mean. Thus the mean is: (4) in 6 ft 7in (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? For 38: / 18 1 below the mean. Thus %. (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%? (b) How many students scored less than a 38? , 816 (d) How many of the 24,000 students receive a scaled score greater than a 90%? Since a 90% corresponds to 1.5 's above: % ,000 1,608 students above 90%. (e) The state would like no more than 550 of the 24,000 students to fail the exam. 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? 2.3% corresponds to 2 's below the mean. Thus the passing score should be set at: % 2ff7e9595c
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