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Statistics

Statistics. Statistics deal with collecting, organizing , and interpreting data. A Survey is a method of collecting information. → Surveys use a small sample to represent a large population . Populations: the whole group ; the group being studied .

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Statistics

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  1. Statistics

  2. Statistics deal with collecting, organizing, and interpreting data. • ASurveyis a method of collecting information. → Surveys use a small sample to represent a large population. • Populations: the whole group; the group being studied. • Sample: part of the population; the group being surveyed.

  3. For each survey topic; determine which represents the population and which represents a sample of the population.

  4. Making Predictions and Drawing Inferences

  5. You can use survey results to predict the actions of a larger group or draw inferences on the entire population. • Predictions: A hypothesis made based on survey results or past actions. • Inference: A prediction that is made using observations, prior knowledge, and experience. • Use proportions to help calculate your predictions and inferences.

  6. A survey found that 6 out of 10 students at IMS have an IPod. Predict how many students have IPods if there are 650 students at IMS. x . 650 6 . 10 cell total = About 390 students have IPods

  7. A researcher catches 60 fish from different locations in a lake. He then tags the fish and puts them back in the lake. Two weeks later, the researcher catches 40 fish from the same locations. 8 of these 40 fish are tagged. Predict the number of fish in the lake. tag total 8 . 40 60 . x = About 300 fish

  8. A middle school has 1,800 students. A random sample of 80 shows that 24 have cell phones. Predict the number of students in the middle school who have cell phones. 24 . 80 x . 1800 phones total = About 540 students have cell phones

  9. A tilapia fish hatchery selectively releases fish when the populations have increased beyond a certain target level. In order to estimate the current fish population, workers at the hatchery catch 110 fish and mark them with special paint. Then a little while later, they catch 530 fish, among which 11 are marked. To the nearest whole number, what is the best estimate for the fish population? 11 . 530 110 . x marked total = About 5,300 fish

  10. In a random sample, 3 of 400 computer chips are found to be defective. Based on the sample, about how many chips out of 100,000 would you expect to be defective? _ x _ . 100,000 3 . 400 defective total = About 750 chips will be defective

  11. Sampling

  12. Biased Sample: A sample that doesn’t truly represent the population. • Example: Surveying 6th graders about the height of IMS students. • Random Sample: A sample where every member of the population has an equal chance of being picked. • Example: Surveying using lockers numbers that end in 2 about the height for IMS students.

  13. Practice ProblemsTell if each sample is biased or random. Explain your answer.

  14. An airline surveys passengers from a flight that is on time to determine if passengers on all flights are satisfied. Biased If they are on-time, they are likely satisfied with their experience.

  15. A newspaper randomly chooses 100 names from its subscriber database and then surveys those subscribers to find if they read the restaurant reviews. Random The names were randomly chosen in such a way that everyone in the population has an equal chance of being picked

  16. The manager of a bookstore sends a survey to 150 customers who were randomly selected from a customer list. Random The customers were randomly chosen so everyone in the population has an equal chance of being picked.

  17. A team of researchers’ surveys 200 people at a multiplex movie theater to find out how much money state residents spend on entertainment. Biased People who go to the movies likely spend more money on entertainment then randomly selected people.

  18. Types of Random Sampling

  19. Simple Random Sample: An unbiased sample where each item or person in the population is as likely to be chosen as any other. • Example: Each students’ name is on a piece of paper in a bowl; names picked without looking • Systematic Random Sample: A sample where the items or people are selected according to a specific time or time interval. • Example: Every 20th person is chosen from an alphabetical list of all students attending IMS.

  20. Stratified Random Sample: A sample where the population is divided into groups; then choose a certain number at random from each group. • Example: Alphabetical list of all students at IMS divided into boys and girls. Then sampling every 20th person from that list.

  21. Types of Biased Sampling

  22. Convenience Sample: A biased sample which consists of members of a population that are easily accessed. • Example: Only surveying one math class about IMS students’ favorite letter day • Voluntary Sample: A biased sample which involves only those who want to participate in the sampling. • Example: Students at IMS who wish to participate in the survey can fill it out on-line

  23. Try the FollowingUse your knowledge of types of Random and Biased Sampling methods to solve the following problems.

  24. To find how much money the average American family spends to cool their home, 100 Alaskan families are surveyed at random. Of the families, 85 said that they spend less than $75 per month on cooling. The researcher concluded that the average American spends less than $75 on cooling per month. Is this conclusion valid? Explain. The conclusion is not valid. This is a biased convenience sample since people in the United States would spend much more than those in Alaska.

  25. Zach is trying to decide which golf course is the best of three golf courses. He randomly surveyed people at a sports store and recorded the results in the table. Which type of sampling method did Zach use? Suppose Zach surveyed 150 more people. How many people would be expected to vote for Rolling green? A simple Random Sample 42 more people

  26. Adults in every 100th household in the phonebook are surveyed about which candidate they plan to vote for. Which type of sampling method is being described? Systematic Random Sample

  27. A computer program selects telephone numbers at random for a survey on which candidate people plan to vote for. Which type of sampling method is being described? Simple Random Sample

  28. The researchers send a mail survey to apple farmers asking them to please record the number of their trees that are infected and send the survey back. Which type of sampling method is being described? Biased – Voluntary Response Sample

  29. To determine what people in California think about a proposed law, 5,000 people from the state are randomly surveyed. Of the people surveyed, 5.8% are against the law. The legislature concludes that the law should not be passed. Which type of sampling method is being described? Is this a valid conclusion? Yes it is valid. A Simple Random Sample was used.

  30. Measures of Central Tendency

  31. Measures of central tendency show what the middle of a data set looks like. • The measures of central tendency are the mean, median, and mode. • The Range is NOT a measure of central tendency

  32. Find the mean, median, mode, and range of the following data set:The ages of Mrs. Long’s grandchildren: 8, 3, 5, 4, 2, 3, 1, and 4.

  33. Meanis average. 1 + 2 + 3 + 3 + 4 + 4 + 5 + 8 = 30 = 3.75 The mean is 3.75

  34. Rangemax minus min Or largest minus smallest. List in order: 1, 2, 3, 3, 4, 4, 5, 8 8 – 1 = 7 The range is 7

  35. Mode the number that occurs most often. There can be several modes or no mode List in order: 1, 2, 3, 3, 4, 4, 5, 8 The mode here is 3 and 4

  36. Medianis the middle data value when in order. The middle two numbers are 3 and 4 List in order: 1, 2, 3, 3, 4, 4, 5, 8 The median is 3.5

  37. Often one measure of Central Tendency is more appropriate for describing a data set. Think about what each measure tells you about the data.

  38. Find the median, mode, mean and range of each data set. Determine the measure of Central Tendency that best describes the data set.

  39. 6, 5, 3, 6, 8 List in order: 3, 5, 6, 6, 8 Median: 6 Mean: 28/5 = 5.6 Mode: 6 Range: 5 Best measure of center: 6 (median & mode)

  40. 7, 6, 13, 16, 15, 9 List in order: 6, 7, 9, 13, 15, 16 Median: 13+9 = 22 22/2 =11 Mean: 66/6 = 11 Range: 10 Mode: none Best measure of center: 11 (median & mean)

  41. 12, 15, 17, 9, 17 List in order: 9, 12, 15, 17, 17 Median: 15 Mean: 70/5 = 14 Range: 8 Mode: 17 Best measure of center: 15 (possibly 14) (median and possibly mean)

  42. 51, 62, 68, 55, 68, 62 List in order: 51, 55, 62, 62, 68, 68 Median: 62 Mean: 366/6 = 61 Range: 17 Mode: 62 & 68 Best measure of center: 62 (median, mean & mode)

  43. List in order: 36, 41, 42, 44, 47 Median: 43 Mean: 210/5 = 42 Range: 11 Mode: none Best measure of center: 42 or 43 (median or mean)

  44. An outlier is an extreme value – either much less than the lowest value or much greater than the highest value.

  45. Use the data set to answer the questions below:4, 6, 3, 6, 25, 3, 2 Is there an outlier?If so, what is it? How does the outlier affect the mean and median? Which measure of central tendency is most effected by an outlier in a data set? Which measure of CT bests describes the data? Explain. List in order: 2, 3, 3, 4, 6, 6, 25 25 Yes With outlier: Median= 4; Mean 49/7 = 7 Without outlier: Median= 3.5; Mean: 24/6 = 4 Mean! Median – it is not dramatically affected by outliers

  46. Types of Graphs First, what is a graph?

  47. Pictographs Histograms Bar Graphs Double Bar Graphs Line Graphs Double Line Graphs Circle (Pie) graphs Line Plots Stem-and-leaf plots Box-and-Whisker Plots Types of Graphs

  48. Pictographs • Use pictures. • What does this graph represent? • How many students play hockey? • 20 • How many more students played soccer than hockey? • 40

  49. Show how often something occurs in equal intervals. This histogram shows: The distance of long jumps at a track meet What range occurred the most? Least? 5’7” – 6’, 6’7” – 7’ How many long jumps were from 5’1” to 6’? 25 long jumps How many more students jumped 5’7” – 6’ than 5’ – 5’6”? 5 students Histograms

  50. Bar Graphs • Use bars of different lengths to display and compare data in specific categories. • This bar graph shows: • The amount of money raised in a charity walk by each of the grades. • Which grade raised twice as much money as 8th? • 10th grade • How much more money did 7th grade raise than 8th ? • $30

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