Statistics. Nominal Ordinal Interval Ratio. Random Sampling Systematic Sampling Stratified Sampling Cluster Sampling

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1 Probability and Statistics CP Final Exam Review Chapter 1 Introduction to Statistics Statistics Descriptive Statistics Inferential Statistics Qualitative Variable Quantitative Variable Discrete Continuous Nominal Ordinal Interval Ratio Random Sampling Systematic Sampling Stratified Sampling Cluster Sampling Survey Observational Study Experiment

2 Examples 1A: Tell whether descriptive or inferential statistics were used. a. The average price of a home sold in Allegheny County during the week of April was $75,328 b. The National Eye Institute has halted a clinical trial on a type of eye surgery, calling it ineffective and possibly harmful to a person s vision. c. Allergy therapy may make bees go away. d. Drinking decaffeinated coffee can raise cholesterol levels by 7%. e. According to the Court Administration of Pennsylvania, 14% of trial-ready civil actions and equity cases in Philadelphia during 1993 were decided in less than six months. f. It is predicted that only 52% of the registered voters will vote in the next election. Examples 1B: Classify each variable as qualitative and quantitative. If quantitative classify as, discrete, or continuous. a. Colors of jackets in a men s clothing store. b. Number of seats in classrooms. c. Classification of children in a day care center (infant, toddler, preschool) d. Length of fish caught in a certain stream. e. Number of students who fail their first statistics test. f. The time it takes to run the 100m Examples 1C: Classify each of the following as nominal, ordinal, interval, or ratio level data. a. Horsepower of motorcycle engines. b. Rating of newscasts in Houston (poor, fair, good, excellent) c. Temperature of automatic popcorn poppers. d. Time required by drivers to complete a course. e. Salaries of cashiers of Day-Night grocery stores. f. Marital status of respondents to a survey on savings accounts. g. Ages of students enrolled in a martial art course. h. Weights of beef cattle fed a special diet. i. Rankings of weight lifters. j. IQ scores of students at Duke University. Examples 1D: Classify each as random, systematic, stratified, or cluster. a. In a large school district, all teachers from two buildings are interviewed to determine whether they believe the students have less homework to do now than in previous years. b. Every seventh customer entering a shopping mall is asked to select his or her favorite store. c. Putting 50 nursing supervisors names in a hat and selecting 10 of them. d. Every 100 th hamburger manufactured is checked to determine its fat content. e. Mail carriers of a large city are divided into four groups according to gender (male or female) and according to whether they walk or ride on their routes. Then 10 are selected from each group and interviewed to determine whether they have been bitten by a dog in the last year. Examples 1E: Classify each as a survey, an observational study, or an experiment. a. You sampled 30 students to determine what their favorite pet was. b. A researcher stood at a busy intersection to see if color of the auto was related to running red lights. c. Subjects are randomly assigned to four groups, each on a different diet to study blood pressures. d. A researcher finds that people who are more hostile have higher cholesterol levels. e. Subjects were randomly assigned to two groups to study respiratory tract infections. f. Mrs. Bloedow asked 90 students what their favorite color was.

3 Chapter 2 Tables and Graph Histogram Graph for: Time-Series Graph Graph for: Pie Graph Compare: Type f % Degrees Stem-and-Leaf Plot Graph for: Box-and-Whisker Plot Graph for: Statistic Parameter Symmetrical Positively Skewed Negatively Skewed

4 Examples 2A: Given the following Histogram find following: Which class has the highest frequency? What is the frequency of class heights ? Which class has the lowest frequency? What is the frequency of class heights ? Examples 2B: Given the following Stem & Leaf Plot find the following: What is the sample size? What is the mode? What is the median? What is the lowest number? What is the highest number? Examples 2C: Given the following Time Series graph find the following: Where does the biggest decrease occur? Over how many years does the graph occur? Example 2D: Given the following Box & Whisker Plot find the following: Highest Number Lowest Number IQR Median Q1 Q3 Example 2E: Give the following distribution, find the following: The number of degrees that would be used to construct a pie graph for Type A blood The percent of those with type B blood The percent of those with type O blood The number of degrees that would be used to construct a pie graph for Type AB blood. Example 2F: Classify each as a statistic or a parameter a. Thirty-six percent of the US adult population has an allergy. b. A sample of 1200 randomly selected adults resulted in 33.2% having an allergy. c. A sample of 300 high school seniors revealed that 85% of them drive. d. Mrs. Bloedow wanted to see how well her students were understanding correlation. She gave all her classes a test and averaged all of their scores.

5 Chapter 3 Measures of Central Tendency, Variation, and Position DATA ARRAY Mean Median Mode Midrange Range Variance Standard Deviation Percentiles Quartiles IQR Outliers UNGROUPED Mean Variance Standard Deviation GROUPED Mean Variance Standard Deviation z-scores

6 Examples 3A: 1. The manager of a sports shop recorded the number of baseball caps he sold during the week The following data represent the total hours spent by college professors instructing and advising students from a study done at the four-year colleges in Ohio. The times reported for the schools were: Calculate the following: a. Mean b. Median c. Mode d. Midrange e. Range f. Variance g. St. deviation h. Q1 i. Q3 j. IQR k. P67 l. P33 m. Outliers? Examples 3B: Find the mean, variance, and standard deviation. 1. The data shown here represent the number of mpg that selected four-wheel drive SUVs obtained in city driving. MPG f The following data show the number of books bought for the semester by CU students in a certain club # Books f Examples 3C: Find the mean, variance, and standard deviation. 1. A sample of snow throwers of a given brand were filled with gasoline (one gallon) and allowed to run until the tank was empty. The times (in minutes) that the snow throwers operated were recorded below in the following distribution. Time f Examples 3D: 2. The following distribution represents the record high temperatures for western states. Temps f What is the z-score for: a. a science grade of 60 on a test with a mean of 75 and a standard deviation of 9 b. a math grade of 78 on a test with a mean of 88 and a standard deviation of 7 2. Which of the following exam grades has a better relative position: a science grade of 65 on a test with a mean of 78 and a standard deviation of 11 or a math grade of 73 on a test with a mean of 85 and a standard deviation of 12?

7 Chapter 4 Correlation and Regression Analysis Interpret r value: Strong Weak Strong Negative Positive Explain why the following data have a correlation coefficient of zero when the data show a very definite pattern. Correlation Analysis: Purpose Result Regression Analysis: Purpose Result - Give an example of the following type of graph: Linear Graph Quadratic Graph Exponential Graph Give an example of the following type of graphs: Positive Correlation Negative Correlation No Correlation Give a real world example of the following type of correlations: Positive/Strong Negative/Strong No Correlation

8 Example 4A: 1. For a group of army inductees, the weight, x, and exercise capacity, y, were recorded for 10 individuals. X Y a. Calculate the correlation coefficient (r). b. Find the EQUATION of the line of best fit. c. Predict the exercise capacity if an inductee weighs 230 pounds. d. The scatter plot would show what kind of trend? 2. A manager wishes to find out whether there is a relationship between the number of radio ads aired per week and the amount of sales (in thousands of dollars) of a product. No. of ads, x Sales, y a. Calculate the correlation coefficient (r). b. Find the EQUATION of the line of best fit. c. Predict the amount of sales if a manager airs 6 ads per week. d. The scatter plot would show what kind of trend? 3. A teachers whishes to determine if there is a relationship between the number of days a student is absent and the student s final grade in the class. # of AB Final Grade a. Calculate the correlation coefficient (r). b. Find the EQUATION of the line of best fit. c. Predict the final grade for a student who is absent 10 days. e. The scatter plot would show what kind of trend?

9 Chapter 5A Probability Part 1 Tree Diagrams: An artist has two choices of paint: acrylics and oil and 5 choices of colors: red, yellow, green, blue, and black. Construct a tree diagram to show all the possible ways she can paint. Counting Rules: Fundamental Counting Rule: A coin is tossed and a die is rolled. Find the number of outcomes. ID Cards: Westside has 5 digit id cards. How many can be created if repetitions are allowed? Westside has 5 digit id cards. How many can be created if repetitions not allowed? Hanna has 4 letter id cards. How many can be created if repetitions are allowed? Hanna has 4 letter id cards. How many can be created if repetitions are not allowed? Factorial Notation: How many different ways can a florist arrange 7 arrangements on a wall? Exponents: Mrs. Bloedow gives a 5 question quiz with 4 choices for each question. How many different keys can be made? A coin is tossed 5 times. How many different ways can this happen? Permutation: Combination:

10 Examples 5aA: 1. If blood types can be A, B, AB, and O and then each type is either RH+ and RH-, draw a tree diagram for the possibilities. 2. A customer can order a sundae with or without whip cream and then with or with out a cherry, draw a tree diagram to determine the possible sundae combinations. 3. The letters A, B, C, and D are placed into a box. A letter is selected and NOT REPLACED. A second number is then drawn. Display the outcomes using a tree diagram. 4. The letters A, B, C, and D are placed into a box. A letter is selected and REPLACED. A second number is then drawn. Display the outcomes using a tree diagram. 5. Mrs. Bloedow would like to have 2 kids. What is the sample space? 6. If you toss 3 coins, what is the sample space. Examples 5aB: 1. Give the number of possible outcomes for the genders of the children in a family of Mrs. Smith givens a 7 question quiz. How many keys can be made if there are 3 choices for each question? 3. A doctor has 5 patients to visit. How many different ways can she make her rounds if she visits each patient only once? 4. How many ways can you arrange 7 different types of soap on a wall. 5. A man has 3 shirts, 5 pants, and 4 ties. How many different outfits can he wear? 6. The numbers 1, 2, 3, 4, 5, 6, and 7 are used to form a five number secret code. Repetitions are not allowed. How many different codes are possible? 7. The numbers 1, 2, 3, 4, 5, 6 and 7 are used to form a five number secret code. Repetitions are allowed. How many different codes are possible? 8. The locker combinations are two letters followed by two numbers. How many different possible combinations are there if repetitions are allowed? 9. The locker combinations are two letters followed by two numbers. How many different possible combinations are there if repetitions are not allowed? 10. If a code has 3 letters followed by 2 digits. How many can be made if repetitions are allowed. 11. If a code has 3 letters followed by 2 digits. How many can be made if repetitions are not allowed. Examples 5aC: 1. How many different ways can Mrs. Bloedow select 3 fruit from a choice of 15 different fruit? 2. If 20 horses are entered into a horse show, in how many ways can the judges award a 1 st, 2 nd, 3 rd, and 4 th place prize? 3. How many different ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available? 4. How many different tests can be made from a test bank of 20 questions if the test consists of 5 questions? 5. In a club there are 6 women and 4 men. A committee of 4 women and 2 men is to be chosen. How many different possibilities are there? 6. How many ways can a dinner patron select 4 salads and 3 veg if there are 6 salads and 5 veg on the menu?

11 Chapter 5B/5C Probability Part 2 & Part 3 Law of Large Number Empirical Probability Theoretical Probability Subjective Probability Mutually Exclusive Independent With Replacement Without Replacement Complement Additional Rule Conditional Probability Examples 5 A: 1. If a die is rolled one time, find these probabilities. a. Of getting a 4. b. Of getting an even number c. Of getting a 9 d. Of getting a number less than 7 e. Of getting a number smaller than 5 f. Of getting a number greater than 4 2. The probability of a storm tomorrow is 6/7. What is the probability of no storm tomorrow? 3. The probability of snow tomorrow is What is the probability of no snow tomorrow? 4. Identify the following as Mutually Exclusive or Not Mutually Exclusive: a. Roll a die: Get an even number and get a number less than 3 b. Roll a die: Get a number greater than 3, and get a number less than 3 c. Select a student in your class: The student has blonde hair, the student has blue eyes d. Select a registered voter: The voter is Republican, the voter is Democrat. 5. Identify the following as Independent or Dependent. a. Drawing a ball from an urn, not replacing it, and then drawing a second ball. b. Drawing a ball from an urn, replacing it, and then drawing a second ball. c. Getting a raise in salary and purchasing a new car. d. Having a large shoe size and having a high IQ. 6. Identify the following as Theoretical / Empirical / Subjective: a. Sam was playing horseshoes and hit the pole 10 out of 50 throws. b. A coin is tossed. The probability of it landing tails up is ½. c. A weather forecaster says that the probability of rain tomorrow is 35%.

12 Examples 5 B: 1. Two cards are drawn from a deck of cards and not replaced. What is the probability of getting 2 Jacks? 2. Three cards are drawn from a deck of cards and replaced. What is the probability of getting 3 spades? 3. In Mrs. Bloedows class there are 30 students 13 of which are made. If four students are selected without replacement, what is the probability of selecting all male? 4. In Statistics there are 5 Sophomores, 9 Juniors, and 15 Seniors. If a student is selected at random, what is the probability that they are a Senior. 5. In Statistics there are 5 Sophomores, 9 Juniors, and 15 Seniors. If a student is selected at random, what is the probability that they are a Junior or a Senior? 6. There is a 76% chance that a man will give a valentine card to his wife. If 4 men are selected at random, what is the probability that all 4 will give valentines to their wives? 7. If a card is selected, what is the probability of getting a king or a heart? 8. If a card is selected, what is the probability of getting a heart and a 4? Examples 5 C: 1. Suppose a certain ophthalmic trait is associated with eye color. One hundred and fifty randomly selected individuals are studied with results as follows: Trait Blue Brown Other Yes No a. Compute the probability that the person has an eye color of other. b. Compute the probability that the patient has the trait or has brown eyes. c. Compute the probability that the patient does not have the trait given they have blue eyes. 2. A medical clinic in Chicago classifies the patients files by gender and by type of diabetes. Type I Type II Male Female a. Compute the probability that a person has Type II diabetes. b. Compute the probability that a person is a male or has type I diabetes. c. Compute he probability that a person has type II diabetes given they are a female. 3. A hospital classifies come of the patients files by gender and by type of care received (ICU or surgical unit). ICU Surgical Male Female a. Compute the probability that a person is a female. b. Compute the probability that a person is a male and is in ICU c. Compute the probability that a person is in the surgical unit and is a female.

13 Chapter 6 Probability Distributions Requirements for a Probability Distribution: Probability Distribution: Binomial Experiment:

14 Examples 6a: Find the missing value in the following probability distribution 1. x P(x) 0.34? x P(x)? x P(x) ? 0.50 Calculate the mean, variance, and standard deviation of the following probability distributions. 4. x P(x) x P(x) A survey found that 33% of Americans have visited a doctor last month. If 10 people are selected at random, find the probability that exactly 5 have visited the doctor last month. 7. A survey found that 43% of the people in a community use the library in one year, If 12 people are selected at random, what is the probability that exactly 7 have visited the library. 8. A survey found that 75% of the people were unaware that maintaining a healthy weight could reduce the risk of stroke. If 12 people are selected, find the probability that at least 10 were unaware that maintaining a healthy weight could reduce the risk. 9. A survey found that20% of the people in a community use the emergency room at a hospital in one year. If 15 people are selected at random, find the probability that no more than 5 use the ER. 10. It was found that 58% of American victims of health care fraud are senior citizens. If 12 victims are randomly selected, find the probably that more than 8 are senior citizens. 11. A study found that 5% of Social Security recipients are too young to vote. If 200 recipients are selected, find the mean and standard deviation. 12. It has been reported that 73% of the federal government employees use . If a sample of 150 federal government employees are selected, find the mean and standard deviation.

15 Chapter 7 Normal Probability Distributions Each month, an American household generates an average of 28 pounds of newspaper garbage or recycling. Assume the standard deviation is 2 pounds. If a household is selected at random, find the probably of generating less than 23 pounds per month? A entry level test has a mean of 400 and a standard deviation of 80. To score in the top 20% what must a person make on the test? A college entrance exam has a mean of 25 and a standard deviation of 4. The bottom 25% must take a special class class to attend the college. What is the score to put a person in the bottom 25%?

16 Examples 7A: 1. Give the following areas: a. Between z = 0 and z = 2.46 b. To the right of z = c. To the left of z = d. Between z = and z = 1.09 e. Between z = and z = f. To the left of z = and to the right of z = Find the following probabilities: a. P(0.91 < z < 1.75) b. P(-1.04 < z < 0) c. P(-0.41 < z < 1.96) d. P(z < 2.04) e. P(z > 1.94) Examples 7B: 1. For a specific year, Americans spend an average of $71.12 for books. Assume the variable is normally distributed. If the standard deviation of the amount spent on book is $8.42, find these probabilities for a randomly selected American. a. He or she spent less than $60 per year on books. b. He or she spent at least $80 per year on books. c. He or she spent between $70 and $75 per year on books. d. He or she spend between $72 and $73 per year on books. 2. To qualify for security officer training, recruits are tested for stress tolerance. The scores are normally distributed with a mean of 85 and a standard deviation of 8. If only the top 15% of recruits are selected, find the cutoff score. 3. The salary s for nurses have an average of $40,000 with a standard deviation of $3500. What is the cutoff salary for a nurse in the bottom 15%? 4. Find the z-score associated with the 95 th Percentile.

17 Chapter 8 & 9 Confidence Intervals & Hypothesis Testing CI n = HT z z z t t p p p Examples 8/9A: 1. A study of 40 bowlers showed that their average score was 186. The standard deviation is 6. Find the 90% confidence interval of the mean score for all bowlers. 2. An automobile shop manager timed six employees and found that the average time it took them to change a water pump was 18 minutes. The standard deviation of the sample was 3 minutes. Find the 99% confidence interval of the true mean. 3. A survey of 15 large US cities finds that the average commute time one way is 25.4 minutes. A chamber of commerce executive feels that the commute in his city is less and wants to publicize this. He randomly selects 25 commuters and finds the average is 22.1 minutes with a standard deviation of 5.3 minutes. Using =0.10, is he correct? 4. How many cities growing season would have to be sampled in order to estimate the true mean growing season with 95% confidence within 2 days. The standard deviation is 54.2 days. 5. A federal report indicated that 27% of children ages 2 to 5 years had a good diet. How large a sample is needed to estimate the true proportion of children with good diets within 2% with 95% confidence. 6. The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24,672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selected a random sample of 35 stroke victims at the hospital and found that the average cost of their rehabilitation is $25,226. The standard deviation is $3251. At =0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $24,672? 7. In a certain state, a survey of 500 workers showed that 45% belonged to a union. Find the 90% confidence interval of the true proportion of workers who belong to a union. 8. An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At =0.05, is there enough evidence to support the attorney s claim?

18 Chapter 10 Inferences Involving Two Populations Examples 10A: 1. Two groups of students are given a problem-solving test, and the results are compared. At α = 0.05, test the claim that math majors score higher than computer science majors. Math Majors Computer Science Majors Hypothesis: n s 1 1 X n s 2 2 X z = p = Reject H o. Fail to reject H o 2. Two brands of batteries are tested and their voltage is compared. At α = 0.05, test the claim that Brand X is last longer than Brand Y. Brand X Brand Y Hypothesis: n s 1 1 X n s 2 2 X z = p = Reject H o. Fail to reject H o