STUDENT MANUAL ALGEBRA II / LESSON 161 Lesson One Hundred and Sixty-One Normal Distribution for some Resolution Today we re going to continue looking at data sets and how they can be represented in different forms. Let s say, for example, that we took a group of students and measured their feet in inches, to the nearest half-inch. We then place this information into a frequency table: Size of feet in inches 6.5 1 7.0 13 7.5 37 8.0 31 8.5 15 9.0 3 Frequency (f) We can take the data in this frequency table and represent it in multiple ways.
STUDENT MANUAL ALGEBRA II / LESSON 161 If we look at the bar graph of this data, we can see that there is almost a bell-like curve in the data set:
STUDENT MANUAL ALGEBRA II / LESSON 161 This type of graph represents what is known as a bell-curve or a normal distribution. In a bell curve, the median values peak the highest, so the values in the middle have the greatest value. The axis of symmetry here, in the middle portion will distributed the curve evenly, 50% on each side. The mean approaches each side of the horizontal axis to create the axis of symmetry. These normal curves can often be used for averages and measurements for manufacturing goods, such as weight of bananas, daily temperature lows, etc. These products and their features are normally distributed. Generally, the features of normal distribution are that 68% of the values in the graph are just 1 standard deviation of the mean. Then, about 95% of the rest of the values will lie within 2 standard deviations of the mean. These assumptions are represented in a normal bell curve. When we think about standard deviation as represented on a normal distribution bell curve, we can think about it as though the axis of symmetry, the point where it is divided in half, is known as the mean. The area that comes between the curve and the axis with the inch values can be represented as one or 1 standard deviation.
STUDENT MANUAL ALGEBRA II / LESSON 161 Here we can see that about 68% of the curve lies between the first standard deviation and -1 and 95% of it is between the standard deviation and -2. For example, let s say that a normal distribution has a mean of 30, and a standard deviation of 6. We can see here that the x axis is divided into 6 sections, meaning that the standard deviation is 6. We want to ask ourselves, what percentage of the values in this graph lie between the 24 and 36? The 2 standard deviations?
STUDENT MANUAL ALGEBRA II / LESSON 161 As we can see, 1 standard deviation below the mean is the value to the left, and here it is 24. One standard deviation, to the right of the mean is 36. Since there are 2 standard deviations between them, we can assume from the generally accepted rule that 68% of the values represented on the curve will be between 24 and 36. If we wanted to determine the probability that an x value, chosen at random would fall between the 42 and 30 value mark, then we can see that 42 is above the mean, to the right, just 2 standard deviations. We know that 95% of the values on the graph are 2 standard deviations from the mean. There are half of them above the mean and half below. Therefore, we divide 95% in half and we get 47.5%. We now know that 47.5% of the values are between 30 and 42. We can represent these values in the following way: P(30 <X < 42) =.475 Finally, let s assume we had 400 values or so. We want to know what percentage of those values would be above the 42. Since the values from 18 to 42 make up 95% of the graph, then we know that there is 5% of the graph left.
STUDENT MANUAL ALGEBRA II / LESSON 161 Each of the values below 18 and above 42 must total 5% of the graph. We can divide that by 2, 1 for each side, and we determine that each remaining portion is worth 2.5% of the total value. So we need to find 2.5% of 400. 400 x.025 = 10. So now we know that about 10 of the values of 400 could be expected to be above 42.
STUDENT MANUAL ALGEBRA II / LESSON 161 Lesson One Hundred and Sixty-One Normal Distribution for some Resolution Name: Date: Now, based on what you ve learned about standard deviation and normal distribution, answer the questions below. Show all your work. 1. If a normal distribution has a standard deviation of 9 with a mean of 45, then what percentage of the values would fall between 36 and 45? Create a bell-curve to represent this data. 2. If a normal distribution has a standard deviation of 9 with a mean of 45, then if you had 100 values, how many of them would you predict would be below 27? Create a bell-curve to represent this data. 3. Let s say that a piggybank contains 200 coins whose weight is normally distributed. How many coins have weights that are 1 standard deviation of the mean? Create a bell-curve to represent this data. 4. Let s say that a piggybank contains 200 coins whose weight is normally distributed. How many of the weights of the coins would be more than 1 standard deviation above the mean?
STUDENT MANUAL ALGEBRA II / LESSON 161 5. If the energy of a light bulb is normally distributed over time and the mean is 45 days with a standard deviation of 3 days. Then what percentage of the light bulb s energy will be distributed less than 42 days? 6. If the daily temperature highs of Las Vegas are normally distributed, and the average temperature is 78 with a standard deviation of 12, then what percentage of the days would be higher than 90? 7. If the scores for a high school football game per player were normally distributed with a mean of 2 and a standard deviation of.6, then find the probability for a randomly selected score of X for P(1.4 < X < 3.2). 8. If the scores for a high school football game per player were normally distributed with a mean of 2 and a standard deviation of.6, then find the probability for a randomly selected score of X for P(0.8 < X < 3.2). 9. If the scores for a high school football game per player were normally distributed with a mean of 2 and a standard deviation of.6, then find the probability for a randomly selected score of X for P(X >0.8).
STUDENT MANUAL ALGEBRA II / LESSON 161 10. A bag contains rubber bands with lengths that are normally distributed with a mean of 6 cm of length, and a standard deviation of 1.5 cm. How many of the rubber bands have lengths within one standard deviation of the mean?
STUDENT MANUAL ALGEBRA II / LESSON 162 Lesson One Hundred and Sixty-Two The Probabilities and Possibilities of Normal Distributions In the previous lesson we learned about bell-curves and normal distributions. Today we re going to continue learning about normal distributions and we re going to look at probabilities for certain event that include normal distributions. Let s say, for example, that the height of a certain group of people is normally distributed. The mean of the data is 67 inches with a standard deviation of 2.5 inches. What would the probability be that the person is less than 6 feet tall? Now first, let s make 6 feet into inches. 6 feet = 72 inches, so 6 feet or 72 inches is 2 standard deviations above the mean. We know that 47.5 \% of the heights X lie between the intervals of 67 and 72, but another 50% lie below 67, so this makes the total 97.5% or P(x < 72) =.975
STUDENT MANUAL ALGEBRA II / LESSON 162 Lesson One Hundred and Sixty-Two The Probabilities and Possibilities of Normal Distributions Name: Date: Now, based on what you ve learned about standard deviation and normal distribution with probabilities, answer the questions below. Show all your work. 1. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that a randomly chosen score is above 200? 2. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it would be between 160 and 240?
STUDENT MANUAL ALGEBRA II / LESSON 162 3. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it s between 120 and 240? 4. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it is below 240? 5. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it above 280?
STUDENT MANUAL ALGEBRA II / LESSON 162 6. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it is more than 1 standards deviation from the mean? 7. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it is 2 standard deviations from the mean? 8. If the average yearly snowfall in Michigan is normally distributed with an average of 39 inches and a standards deviation of 9 inches, then what is the probability, in percentages, that a random year will have more than 30 inches of rain?
STUDENT MANUAL ALGEBRA II / LESSON 162 9. Cracker boxes say that there is 12 oz. in them, but the total weight of the crackers is normally distributed, and the mean is 12.3 oz. The standard deviation is.3 oz. What percentage of cracker boxes will probably contain fewer than 11.7 oz. of crackers? 10. A single serving coffee maker fills up cups with an average of 8 oz. The standard deviation is.4 oz. If there were 50 cups of coffee served, then how many of those cups would you expect to be filled with.4 oz. of the mean with normal distribution?
STUDENT MANUAL ALGEBRA II / LESSON 163 Lesson One Hundred and Sixty-Three The Graph s Turn to Stand in Line Random samplings can be demonstrated in many ways, as seen in the previous lessons. Some graphs are better to use for certain types of data sets than others. For example, if we measured the test scores of students and compared those average scores to the amount of time students spent studying, then we might create a data set like this: Hours Studied Score 1 12 2 36 4 60 6 66 7 96 It wouldn t really help us to see this information in a pie graph because the information here does not equal 100% of something. It is a random sampling. For this type of assosiated study, meaning that each of the outcomes are conditional to the amount of time spent studying, then it s best to use a line graph, or a linear representation of the data. As you can see in the graph, the equation of this line is in the slope-intercept form of y = mx + b or y = 19.8x 5.4.
STUDENT MANUAL ALGEBRA II / LESSON 163 Remember from previous lessons that m is the slope and b is the y-intercept, or the point on the graph that where the line crosses the y-axis. Let s say we calculated the average number of calories burned per mile ran. Perhaps we record the data, and it ends up looking like this: Miles Ran Calories Burned 1 105 2 209 3 314 4 419 5 525 This is an associated study, so the number of calories burned depends on how far the runner ran. We can plot these x and y coordinate points on a graph like so: We can use any two x and y points on the graph to determine the slope: (3, 314) and (4, 419) m = y 2 y 1 (This is the change in the points, how much is goes up or rises) x 2 - x 1 (This is the change in the x points, how much is goes over or runs) 209 105 2-1 104 1 = 104
STUDENT MANUAL ALGEBRA II / LESSON 163 Now, we can use our x and y coordinates, and then plug in to solve for the y intercept: 105 =104 + b b = 1 The approximate equation for this line is y = 104x + 1
STUDENT MANUAL ALGEBRA II / LESSON 163 Lesson One Hundred and Sixty-Three The Graph s Turn to Stand in Line Name: Date: Now, based on what you ve learned about linear functions, complete the practice problems below. 1. Plot the following set of data and create a line graph: 1 8 2 16 3 24 4 32 5 40 6 48 2. What is the slope of the following linear function created by the data table? 1 8 2 16 3 24 4 32 5 40 6 48
STUDENT MANUAL ALGEBRA II / LESSON 163 3. Mark tracked how many pages he read every ten minutes. He ended up read approximately 15 pages per every 10 minutes of reading. Make a table to represent the results after 50 minutes: 4. Why wouldn t it be a good idea to create a pie graph to represent this data? 5. Now, create a linear graph to represent the data: 6. What is the slope of the graph?
STUDENT MANUAL ALGEBRA II / LESSON 163 7. Mr. Powers had a stack of check to sign for his employees. After 25 minutes he had completed 30, then he began to record how many he could complete. He found that every 10 minutes, he could sign 20 more than the initial 30 he had when he started recording. Make a table to represent this data up to 65 minutes. 8. Now, plot the points on a graph: 9. What is the slope of the line? 10. Write an equation for this line:
STUDENT MANUAL ALGEBRA II / LESSON 164 Lesson One Hundred and Sixty-Four Predicting Populations Environmental engineers often need to do various studies about population growths of animals or insects in certain areas. It s very difficult for them to make exact accountings, so they use samplings to help them make the predictions. Let s say, for example that in a specific wetland, 100 frogs have been caught, marked, and then returned back into the marsh. A month later, they took a sampling of 100 more frogs, and out of the 100, only 2 of them had been previously marked. They can now use this data to make predictions about the population of this species of frog in the area. They use the ratio of the marked frogs from the second sampling as the determinate for finding the population of the total amount of frogs in the area. So 2 frogs represent 100 frogs (from the first sampling), and 100 frogs is 2% of the total population (from the second sampling). We can track the data like so: 100/n = (the total number marked, over the number in the population n) is equal to 2/100 or the number marked in the second sample over the total number in the second sample). 100/n = 2/100 We can solve for n: n = 100 100/ 2 n = 5000 The total estimated population of frogs in that area is 5000.
STUDENT MANUAL ALGEBRA II / LESSON 164 Lesson One Hundred and Sixty-Four Predicting Populations Name: Date: Now, based on what you ve learned about linear functions, complete the practice problems below. A team of environmental engineers captures and marks 200 rabbits in a specific region. The rabbits are released back into the environment and mix with the others. On the second sampling, the team returns and observes that 5 of the first 40 rabbits caught were marked from the original sampling. 1. What is the total assumed population of the rabbits in that region? 2. What would the population be if only 100 rabbits had been initially marked.
STUDENT MANUAL ALGEBRA II / LESSON 164 3. What would the population be if the team had marked 100 rabbits and they only spotted 1 marked out of 20 upon the second visit. 4. What would the rabbit population be if 60 had been marked initially and the team spotted 3 marked deer out of 50 later on. 5. What would be the population estimate if they had caught 150 rabbits each time, and on the second catch they only found 5 originally marked rabbits?
STUDENT MANUAL ALGEBRA II / LESSON 164 Naturalists studying specific regions of the rainforest wanted to estimate the population of butterflies in a certain area. In the first sampling they caught and marked 75 butterflies, in the second sampling, they also collected 75 more butterflies and found that only 2 of them had been originally marked. 6. What is the estimated butterfly population based on the findings of the two samplings? 7. What would the population estimate be if they had caught 150 butterflies the second time and only 6 of them had been originally marked from the original sampling of 75? 8. What would the population estimate be if they had caught 250 butterflies the second time and only 10 of them had been originally marked from the original sampling of 75?
STUDENT MANUAL ALGEBRA II / LESSON 164 9. What would the population estimate be if they had caught 30 butterflies the second time and only 3 of them had been originally marked from an original sampling of 25? 10. What would the population estimate be if they had caught 300 butterflies the second time and only 15 of them had been originally marked from an original sampling of 250?
STUDENT MANUAL ALGEBRA II / LESSON 167 Lesson One Hundred and Sixty-Seven Statistics and Probability Cumulative Review Part 1 Name: Date: Please complete the following Part 1 cumulative review; you will complete the second part of this review in the next lesson. Ask your instructor if you have any questions. You may want to review some of the instructional sections of previous lessons about Statistics and Probability to help you remember certain processes as you complete the work below. These will be the same concepts covered in the upcoming assessments. SHOW ALL YOUR WORK: 1. A bag contains rubber bands with lengths that are normally distributed with a mean of 6 cm of length, and a standard deviation of 1.5 cm. How many lengths are more than 1 standard deviation above the mean? 2. A bag contains rubber bands with lengths that are normally distributed with a mean of 6 cm of length, and a standard deviation of 1.5 cm. Wheat percentage of rubber bands can be expected to be longer than 7.5 cm?
STUDENT MANUAL ALGEBRA II / LESSON 167 3. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it is more than 1 standards deviation from the mean? 4. If students scores were normally distributed and the mean was 200 with a standard deviation of 40, then what is the probability, in percentages, that it is 2 standard deviations from the mean? 5. If the average yearly snowfall in Michigan is normally distributed with an average of 39 inches and a standards deviation of 9 inches, then what is the probability, in percentages, that a random year will have more than 30 inches of rain? 6. Gerald started tracking how many words per minute he could type. He realized that he could type 60 words a minute. Create a table to represent this data up to 5 minutes.
STUDENT MANUAL ALGEBRA II / LESSON 167 7. Now, plot the points on a graph: 8. Is this data conditional or random? Why? Explain. 9. What is the slope of the line? 10. Write an equation for this line:
STUDENT MANUAL ALGEBRA II / LESSON 167 Naturalists studying specific regions of the forest wanted to estimate the population of caterpillars in a certain area. In the first sampling they caught and marked 75 caterpillars, in the second sampling, they also collected 75 more caterpillars and found that only 2 of them had been originally marked. 11. What is the estimated caterpillars population based on the findings of the two samplings? 12. What would the population estimate be if they had caught 150 caterpillars the second time and only 6 of them had been originally marked from the original sampling of 75? 13. What would the population estimate be if they had caught 250 caterpillars the second time and only 10 of them had been originally marked from the original sampling of 85?
STUDENT MANUAL ALGEBRA II / LESSON 167 14. What would the population estimate be if they had caught 30 caterpillars the second time and only 3 of them had been originally marked from an original sampling of 25? 15. What would the population estimate be if they had caught 400 caterpillars the second time and only 15 of them had been originally marked from an original sampling of 250? Let s say you have a deck of cards, what is the probability that the following will occur? 16. You draw a card of the suit of Spades:
STUDENT MANUAL ALGEBRA II / LESSON 167 Based on the Standard American English alphabet, what is the probability that of the following would occur by choosing a letter at random? 17. You choose A & B: Based on a six-sided dice, what is the probability that of the following would occur by rolling it? 18. You roll a 4: 19. Let s say a coin is tossed 4 times. Create a sample set that demonstrates the 16 different results. You might find it helpful to create a tree diagram to find the possible results:
STUDENT MANUAL ALGEBRA II / LESSON 167 Based on the information from the sample space above in question #7, answer the following questions. What is the probability that the four tosses will result in 20. At least two heads? 21. With three heads at the most? Determine if the following samples are or are not mutually exclusive. If they are mutually exclusive write mutually exclusive if they aren t mutually exclusive, then write not mutually exclusive. 22. A girl; sister: 23. A grandmother; a father
STUDENT MANUAL ALGEBRA II / LESSON 167 A circle is divided into 4 parts: A, B, C, D 24. If the pointer is spun repeatedly, then how many different results could occur from 2 spins? 25. If the pointer is spun repeatedly, then how many different results could occur from 8 spins? In how many different ways can each of the letters in the following words be arranged? Show your work and solutions. 26. HIM 27. HERS
STUDENT MANUAL ALGEBRA II / LESSON 167 Find the number of permutations of the digits or letters below: 28. SITTER: 29. 1377: 30. 25755:
STUDENT MANUAL ALGEBRA II / LESSON 168 Lesson One Hundred and Sixty-Eight Statistics and Probability Cumulative Review Part 2 Name: Date: Please complete the following Part 2 cumulative review; you should have completed part one in the previous lesson. Ask your instructor if you have any questions. You may want to review some of the instructional sections of previous lessons about Statistics and Probability to help you remember certain processes as you complete the work below. These will be the same concepts covered in the upcoming assessments. SHOW ALL YOUR WORK: 1. How many different ways can you make combinations of 2 letters out of the four-letter word LIFE? 2. How many different ways can you make combinations of 7 numbers out of the numbers 1 through 7? 3. How many diagonals can be drawn from each vertex of a hexagon?
STUDENT MANUAL ALGEBRA II / LESSON 168 4. How many diagonals can be drawn from each vertex of a pentagon? Answer the following questions about random probabilities. Show all your work, using the combination formula and probability formulas. 5. If there are five different single digits, 1, 2, 3, 4, and 5, then what is the probability that of three numbers, chosen at random, none of them would be 5? 6. 10 different boys are auditioning for a play. The directors are looking to cast only three of the boys. If three of the boys were chosen at random, then what is the probability that all three of the boys chosen are the brothers? Show your answer in fraction and percent, round to the nearest thousandth.
STUDENT MANUAL ALGEBRA II / LESSON 168 If a card is pulled from a deck of playing cards and a six-sided die is tossed, then what is the probability that the following two independent events will occur: 7. Rolling an odd number and drawing a King. 8. Drawing a 10, and rolling an odd number. If a dice is rolled and the spinner below with 5 sides, each with the letters A, B, C, D, E is spun, then what is the probability of the following independent events occurring at the same time? 9. Spinning a vowel.
STUDENT MANUAL ALGEBRA II / LESSON 168 10. Spinning a vowel and rolling an even number. Identical dials are spun, each dial contains X, Y, and Z. What is the probability that: 11. 4 identical spinners would all land on either Y or Z? 12. 5 identical spinners would all land on X?
STUDENT MANUAL ALGEBRA II / LESSON 168 A dealer holds a shuffled deck of 52 playing cards and draws cards at random. Determine the probably that the following events would occur, without replacement. 13. The first card drawn is a spade. 14. The first card dealt is the ace of spades. 15. 6 different numbers are drawn on separately on 11 different cards, one number on each card. There is one 1, three 2 s, two 3 s, three 4 s, one 5, and one 6. 2 cards are drawn in a row with replacement. What is the probability that neither of them are 4 s? 16. Let s say that a piece of dice is rolled four times, and we are looking for how many times a 5 will be rolled. Determine the probability p that a 5 will be rolled.
STUDENT MANUAL ALGEBRA II / LESSON 168 17. Let s say that a piece of dice is rolled four times, and we are looking for how many times a 5 will be rolled. Determine the probability q that anything beside a 5 will be rolled. 18. Let s say that a piece of dice is rolled seven times. Write the expansion in the binomial distribution form to represent the possibility of rolling exactly five 3 s. 19. Find P(3) if n=5 and p = 1/5 20. Find the following values for p, q, x, and n, based on P(4)= 6 C 4 p 4 q 2 when n is equal or greater than x.
STUDENT MANUAL ALGEBRA II / LESSON 168 Refer to this set of data for the following questions: 21. Which column represents the domain? 22. Which column represents the range? 23. How many coins were flipped during this experiment, based on the data given? 24. How many times total were the group of coins flipped? 25. Which number of heads has the greatest frequency? 26. Which number of heads has the lowest frequency? 27. Which values in the data set represent the y-axis values?
STUDENT MANUAL ALGEBRA II / LESSON 168 28. Determine the range and mean for the following set of numbers: 140, 112, 132, 166, 153, 110. 29. Determine the range and mean for the following set of numbers: 8, 8, 7, 5, 2. 30. Find the mean and standard deviation for the following set of numbers: 12, 7, 13, 8.