7th Grade Math Inferences & Probability
PROBABILITY Sampling Click on a topic to go to that section. Comparing Two Populations Introduction to Probability Experimental and Theoretical Word Problems Fundamental Counting Principle Permutations and Combinations Probability of Compound Events Probabilities of Mutually Exclusive and Overlapping Events Complementary Events Common Core: 7.SP.1-8
Sampling Return to table of contents
Your task is to count the number of whales in the ocean or the number of squirrels in a park. How could you do this? What problems might you face? A sample is used to make a prediction about an event or gain information about a population. A whole group is called a POPULATION. A part of a group is called a SAMPLE.
A sample is considered random (or unbiased) when every possible sample of the same size has an equal chance of being selected. If a sample is biased, then information obtained from it may not be reliable. Example: To find out how many people in New York feel about mass transit, people at a train station are asked their opinion. Is this situation representative of the general population? No. The sample only includes people who take the train and does not include people who may walk, drive, or bike.
Determine whether the situation would produce a random sample. You want to find out about music preferences of people living in your area. You and your friends survey every tenth person who enters the mall nearest you.
1 Food services at your school wants to increase the number of students who eat hot lunch in the cafeteria. They conduct a survey by asking the first 20 students that enter the cafeteria to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. A Yes B No
2 The guidance counselors want to organize a career day. They will survey all students whose ID numbers end in a 7 about their grades and career counseling needs. Would this situation produce a random sample? Explain your answer. A Yes B No
3 The local newspaper wants to run an article about reading habits in your town. They conduct a survey by asking people in the town library about the number of magazines to which they subscribe. Would this produce a random sample? Explain your answer. A Yes B No
How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number of wolves on a mountain?
One way to estimate the number of wolves on a mountain is to use the CAPTURE - RECAPTURE METHOD.
They wait until all the wolves have mixed together. Then they find a second group of wolves and count how many are tagged.
Biologists use a proportion to estimate the total number of wolves on the mountain: tagged wolves on mountain tagged wolves in second group total wolves on mountain total wolves in second group For accuracy, they will often conduct more than one recapture. = 8 2 w 9 2w = 72 w = 36 = There are 36 wolves on the mountain
Try This: Biologists are trying to determine how many fish are in the Rancocas Creek. They capture 27 fish, tag them and release them back into the Creek. 3 weeks later, they catch 45 fish. 7 of them are tagged. How many fish are in the creek? There are 174 fish in the river 27 7 f 45 27(45) = 7f 1215 = 7f 173.57 = f =
A whole group is called a POPULATION. A part of a group is called a SAMPLE. When biologists study a group of wolves, they are choosing a sample. The population is all the wolves on the mountain. Population Sample
Try This: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500?
4 860 out of 4,000 people surveyed watched Dancing with the Stars. How many people in the US watched if there are 93.1 million people?
5 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?
6 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.
7 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect?
8 The chart shows the number of people wearing different types of shoes in Mr. Thomas' English class. Suppose that there are 300 students in the cafeteria. Predict how many would be wearing high-top sneakers. Explain your reasoning. Shoes Number of Students
Multiple Samples The student council wanted to determine which lunch was the most popular among their students. They conducted surveys on two random samples of 100 students. Make at least two inferences based on the results. • Most students prefer pizza. • More people prefer pizza than hamburgers and tacos combined.
Try This! The NJ DOT (Department of Transportation) used two random samples to collect information about NJ drivers. The table below shows what type of vehicles were being driven. Make at least two inferences based on the results of the data.
The student council would like to sell potato chips at the next basketball game to raise money. They surveyed some students to figure out how many packages of each type of potato chip they would need to buy. For home games, the expected attendance is approximately 250 spectators. Use the chart to answer the following questions.
9 How many students participated in each survey?
10 According to the two random samples, which flavor potato chip should the student council purchase the most of? A Regular B BBQ C Cheddar
11 Use the first random sample to evaluate the number of packages of cheddar potato chips the student council should purchase.
Comparing Two Populations Return to table of contents
Measures of Variation - Vocabulary Review Range - The difference between the greatest data value and the least data value Quartiles - are the values that divide the data in four equal parts. Lower (1st) Quartile (Q1) - The median of the lower half of the data. Upper (3rd) Quartile (Q3) - The median of the upper half of the data. Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1) Mean absolute deviation - the average distance between each data value and the mean.
Example: Victor wants to compare the mean height of the players on his favorite basketball and soccer teams. He thinks the mean height of the players on the basketball team will be greater but does not know how much greater. He also wonders if the variability of heights of the athletes is related to the sport they play. He thinks that there will be a greater variability in the heights of soccer players as compare to basketball players. He uses the rosters and player statistics from the team websites to generate the following lists. Height of Soccer Players (inches) 73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67, 70, 72, 69, 78, 73, 76, 69 Height of Basketball Players (inches) 75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84 from http://katm.org/wp/wp-content/uploads/flipbooks/7th_FlipBookEdited21.pdf
Victor creates two dot plots on the same scale. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 80 85 65 75 70 Height of Soccer Players (inches) x x x x x x x x x x x x x x x x 80 85 75 65 70 Height of Basketball Players (inches)
Victor notices that although generally the basketball players are taller, there is an overlap between the two data sets. Both teams have players that are between 73 and 78 inches tall. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 80 85 65 70 75 Height of Soccer Players (inches) x x x x x x x x x x x x x x x x 80 85 65 75 70 Height of Basketball Players (inches)
To express the difference between centers of two data sets as a multiple of a measure of variability, first find the difference between the centers. *Recall: The difference between the means is 79.75 – 72.07 = 7.68. Divide the difference by the mean absolute deviations of each data set. 7.68 ÷ 2.14 = 3.59 7.68 2.5 = 3.07 The difference of the means (7.68) is approximately 3 times the mean absolute deviations.
Use the following data to answer the next set of questions. Pages per Chapter in Hunger Games x x x x x x x x x x x x x x x x x x x x x x x x x x x 20 25 30 15 10 Pages per Chapter in Twilight x x x x x x x x x x x x x x x x x x x x x x 20 25 30 15 10
12 What is the mean number of pages per chapter in the Hunger Games?
13 What is the mean number of pages per chapter in Twilight?
14 What is the difference of the means?
15 What is the mean absolute deviation of the data set for Hunger Games? (Hint: Round mean to the nearest ones.)
16 What is the mean absolute deviation of the data set for Twilight? (Hint: Round mean to the nearest ones.)
17 Which book has more variability in the number of pages per chapter? A Hunger Games B Twilight
18 The difference of the means between the two data sets is approximately ______ times the mean absolute deviation for Twilight? (Round your answers to the nearest tenths.)