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**AP STATISTICS EXAM REVIEW by DAVID CUSTER (click on topic of choice)**

AP STATISTICS EXAM REVIEW by DAVID CUSTER (click on topic of choice) TOPIC I: Describing Data (15 questions) TOPIC II: Experimental Design (15 questions) TOPIC III: Probability (18 questions) TOPIC IV: Inference (15 questions)

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**TOPIC I: Describing Data**

Univariate Data Normal Distributions Bivariate Data START TOPIC Topic I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 back to main

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**Which observation has the higher z-score. I. x=25. 4; μ =12. 9; σ=3**

Which observation has the higher z-score? I. x=25.4; μ =12.9; σ=3.7 II. x=25.4; μ=15.3; σ=2.7 I.1. I II z-scores are equal cannot be determined since we don’t know the standard deviations of the populations cannot be determined since we don’t know if the populations are normal Topic I Menu

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**The z-score for I is 3.38; the z-score for II is 3.74**

SOLUTION Which observation has the higher z-score? I. x=25.4; μ =12.9; σ=3.7 II. x=25.4; μ=15.3; σ=2.7 I.1. The z-score for I is 3.38; the z-score for II is 3.74 I II z-scores are equal cannot be determined since we don’t know the standard deviations of the populations cannot be determined since we don’t know if the populations are normal Topic I Menu

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I.2. The Quartile and Percentile positions of the value 10 in the set {10, 6, 8, 9, 12, 17, 32, 16} are: Q3; 38 Q2; 38 Q2; 26 Q3; 26 none of these Topic I Menu

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SOLUTION I.2. The Quartile and Percentile positions of the value 10 in the set {10, 6, 8, 9, 12, 17, 32, 16} are: When the values are in order, there are 3 values below 10. This puts 10 in the 37.5 percentile and the 2nd quartile. Q3; 38 Q2; 38 Q2; 26 Q3; 26 none of these Topic I Menu

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**The Standard Deviation of the set {5, 7, 7, 8, 10, 11} is 2**

The Standard Deviation of the set {5, 7, 7, 8, 10, 11} is 2. Which of the following sets also has a Standard Deviation of 2? I.3. {4, 5, 8, 12, 14} {2, 4, 6, 8, 10, 12} {3, 5, 5, 6, 8, 9} {10, 14, 14, 16, 20, 22} none of the above Topic I Menu

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SOLUTION The Standard Deviation of the set {5, 7, 7, 8, 10, 11} is 2. Which of the following sets also has a Standard Deviation of 2? I.3. There is a uniform decrease of 2 units in this set. The st. deviation must be the same. {4, 5, 8, 12, 14} {2, 4, 6, 8, 10, 12} {3, 5, 5, 6, 8, 9} {10, 14, 14, 16, 20, 22} none of the above Topic I Menu

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**12% still lies between a and b 12% lies between a + d and b + d **

If 12% of the values of a data set lie between a and b and d is added to each value, then which of the following is true? I.4. 12% still lies between a and b 12% lies between a + d and b + d (12+d)% lies between a and b (12+d)% lies between a + d and b + d there is no way to tell how much data is between a and b Topic I Menu

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SOLUTION If 12% of the values of a data set lie between a and b and d is added to each value, then which of the following is true? I.4. A uniform shift of all the data maintains the percentages of data in shifted intervals. 12% still lies between a and b 12% lies between a + d and b + d (12+d)% lies between a and b (12+d)% lies between a + d and b + d there is no way to tell how much data is between a and b Topic I Menu

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I.5. If a distribution is relatively symmetric and mount-shaped, order the following (from least to greatest) a z-score of the value of Q a value in the 70th percentile 1, 2, 3 1, 3, 2 3, 2, 1 3, 1, 2 2, 3, 1 Topic I Menu

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SOLUTION I.5. If a distribution is relatively symmetric and mount-shaped, order the following (from least to greatest) a z-score of the value of Q a value in the 70th percentile 1, 2, 3 1, 3, 2 3, 2, 1 3, 1, 2 2, 3, 1 The percentile of a z-score of 1 is about 84%, and the percentile of Q3 is 75% Topic I Menu

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Which of the following would NOT be a correct interpretation of a correlation coefficient of r = -.30 I.6. The variables are inversely related The coefficient of determination is 0.09 30% of the variation between the variables is linear There exists a weak relationship between the variables All are correct Topic I Menu

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**The value of r2 explains the variation between the variables. Not r.**

SOLUTION Which of the following would NOT be a correct interpretation of a correlation coefficient of r = -.30 I.6. The value of r2 explains the variation between the variables. Not r. The variables are inversely related The coefficient of determination is 0.09 30% of the variation between the variables is linear There exists a weak relationship between the variables All are correct Topic I Menu

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**Which of the following displays is best suited for categorical data?**

I.7. Box Plot Bar Graph Stem and Leaf Plot Dot Plot Scatterplot Topic I Menu

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**Which of the following displays is best suited for categorical data?**

SOLUTION Which of the following displays is best suited for categorical data? I.7. In a bar graph, each column is separate, allowing for categorical separation. Box Plot Bar Graph Stem and Leaf Plot Dot Plot Scatterplot Topic I Menu

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I.8. Linear regression usually employs the method of least squares. Which of the following is the quantity that is minimized by the least squares process? Topic I Menu

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**Least Squares Regression minimizes the residuals in the y-direction.**

SOLUTION I.8. Linear regression usually employs the method of least squares. Which of the following is the quantity that is minimized by the least squares process? Least Squares Regression minimizes the residuals in the y-direction. Topic I Menu

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**Which of the following is NOT true?**

I.9. Two sets of data can have the same means but different variances Two sets of data can have the same variance but different means Two different values in a data set can have the same z-score All the absolute values of z-scores for a data set can be equal All of the above are true Topic I Menu

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**Which of the following is NOT true?**

SOLUTION Which of the following is NOT true? I.9. Since each value is a distinct distance from the mean, the z-scores must all be different Two sets of data can have the same means but different variances Two sets of data can have the same variance but different means Two different values in a data set can have the same z-score All the absolute values of z-scores for a data set can be equal All of the above are true Topic I Menu

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**In a symmetric, mount-shaped distribution, what percentile has a z-score of -2?**

I.10. Topic I Menu

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SOLUTION In a symmetric, mount-shaped distribution, what percentile has a z-score of -2? I.10. 2.5th percentile We should be able to approximate this with the normal distribution. Area to the left of -2? On the TI-83: Normalcdf(-1E99, -2) = Topic I Menu

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**Lee’s z-score on his math test was 1. 5. The class average was a 62**

Lee’s z-score on his math test was The class average was a 62.1 and the variance was What was Lee’s actual grade on the test? I.11. 60 62 64 66 68 Topic I Menu

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**We need to solve the following equation for x:**

SOLUTION Lee’s z-score on his math test was The class average was a 62.1 and the variance was What was Lee’s actual grade on the test? I.11. We need to solve the following equation for x: so x=66 60 62 64 66 68 Topic I Menu

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I.12. Does the following problem have a unique solution? If so, find it. If not, show at least two answers: 5 numbers have Q1=12, Median=15, Q3=18. Find the mean. Topic I Menu

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SOLUTION I.12. Does the following problem have a unique solution? If so, find it. If not, show at least two answers: 5 numbers have Q1=12, Median=15, Q3=18. Find the mean. {11, 13, 15, 17, 19} μ=15 { 9, 15, 15, 17, 19} μ=15 the data sets are not unique, but the mean is always 15! YES!!! surprisingly! Topic I Menu

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I.13. The average grade on a math test given to two sections is Section I has 27 students with a mean grade of If the mean grade of Section 2 is 65.30, how many students are in section 2? Topic I Menu

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SOLUTION I.13. The average grade on a math test given to two sections is Section I has 27 students with a mean grade of If the mean grade of Section 2 is 65.30, how many students are in section 2? 23 students. we arrive at n=23 Topic I Menu

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**If the mean of 75 values is 52. 6 and the mean of 25 values is 48**

If the mean of 75 values is 52.6 and the mean of 25 values is 48.4; find the mean of all 100 values. I.14. 51.55 52.76 56.55 56.88 59.12 Topic I Menu

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SOLUTION If the mean of 75 values is 52.6 and the mean of 25 values is 48.4; find the mean of all 100 values. I.14. 51.55 52.76 56.55 56.88 59.12 75(52.6) + 25(48.4) 100 Topic I Menu

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In skewed-right distributions, what is most frequently the relationship of the mean, median, and mode? I.15. mean > median > mode median > mean > mode mode > median > mean mode > mean > median mean > mode > median Topic I Menu

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SOLUTION In skewed-right distributions, what is most frequently the relationship of the mean, median, and mode? I.15. The median is resistant, the mean, not at all. So a right skewed distribution will have a mean much higher than median, much higher than mode. mean > median > mode median > mean > mode mode > median > mean mode > mean > median mean > mode > median Topic I Menu

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**TOPIC II: Experimental Design**

Sampling Designing Experiments Observational Studies START TOPIC Topic II 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 back to main

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II.1. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. The study is categorized as: a census a survey sample an observational study a designed experiment none of these Topic II Menu

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SOLUTION II.1. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. The study is categorized as: The director is observing behavior, not implementing treatments on the group a census a survey sample an observational study a designed experiment none of these Topic II Menu

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II.2. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director only looks at those in one department, she is performing: a simple random sample a quota sample a convenience sample a multi-stage cluster sample a census Topic II Menu

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SOLUTION II.2. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director only looks at those in one department, she is performing: Without a properly randomized selection, she is introducing bias to the study. a simple random sample a quota sample a convenience sample a multi-stage cluster sample a census Topic II Menu

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II.3. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director selects 50 employees at random and categorizes by gender, she is: blocking for gender testing for a lurking variable promoting sexual harassment testing for bias none of these Topic II Menu

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**She is investigating whether gender affects lunchtime behavior**

SOLUTION II.3. A personnel director studied the eating habits of employees by watching a group of employees at lunch. He wishes to see who buys in the cafeteria, who brings a home lunch, and who goes out. If the director selects 50 employees at random and categorizes by gender, she is: blocking for gender testing for a lurking variable promoting sexual harassment testing for bias none of these She is investigating whether gender affects lunchtime behavior Topic II Menu

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**Which of the following is NOT a concern in data collection?**

II.4. lurking variables blocking bias non-response all of the above are concerns Topic II Menu

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**Which of the following is NOT a concern in data collection?**

SOLUTION Which of the following is NOT a concern in data collection? II.4. Even blocking methods need to be analyzed lurking variables blocking bias non-response all of the above are concerns Topic II Menu

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**Which of the following is NOT a valid sample design?**

II. 5. Code every member of a population and select 100 randomly chosen members Divide a population by gender and select 50 individuals randomly from each group Select individuals randomly and place into gender groups until you have the same proportion as in the population Select five homerooms at random from all the homerooms in a large high school. All of these are valid Topic II Menu

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**Which of the following is NOT a valid sample design?**

SOLUTION Which of the following is NOT a valid sample design? II. 5. But you may be able to question the validity of answer (C) Code every member of a population and select 100 randomly chosen members Divide a population by gender and select 50 individuals randomly from each group Select individuals randomly and place into gender groups until you have the same proportion as in the population Select five homerooms at random from all the homerooms in a large high school. All of these are valid Topic II Menu

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An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The variable of interest is: II.6. the doctor’s specialty, e.g. cardiology, obstetrics, etc. the number of doctors who are cardiologists all cardiologists in the American Medical Association directory a random sample of 100 cardiologists none of these Topic II Menu

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**we are interested in the percentage of doctors who have been sued**

SOLUTION An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The variable of interest is: II.6. the doctor’s specialty, e.g. cardiology, obstetrics, etc. the number of doctors who are cardiologists all cardiologists in the American Medical Association directory a random sample of 100 cardiologists none of these we are interested in the percentage of doctors who have been sued Topic II Menu

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An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The population of interest is: II.7. the set of all doctors who were sued for malpractice the set of cardiologists who were sued for malpractice all doctors all cardiologists all doctors who have malpractice insurance Topic II Menu

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**Just cardiologists, not all doctors.**

SOLUTION An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. The population of interest is: II.7. the set of all doctors who were sued for malpractice the set of cardiologists who were sued for malpractice all doctors all cardiologists all doctors who have malpractice insurance Just cardiologists, not all doctors. Topic II Menu

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An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. Which could be used to gather the data? II.8. a designed experiment a census of all cardiologists an observational study of randomly selected cardiologists a survey sent to randomly selected cardiologists any answer except (A) Topic II Menu

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SOLUTION An insurance company conducted a study to determine the percent of cardiologists who had been sued over the last 5 yrs. Which could be used to gather the data? II.8. This population of this observational study is too large to track everyone down. a designed experiment a census of all cardiologists an observational study of randomly selected cardiologists a survey sent to randomly selected cardiologists any answer except (A) Topic II Menu

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**Which of the following is NOT a source of bias in sample surveys?**

II.9. non-response wording of questions voluntary response use of a telephone survey all are sources of bias Topic II Menu

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**Which of the following is NOT a source of bias in sample surveys?**

SOLUTION Which of the following is NOT a source of bias in sample surveys? II.9. and don’t forget… even a huge sample size can’t correct a poorly selected sample. Remember the Literary Digest Poll! non-response wording of questions voluntary response use of a telephone survey all are sources of bias Topic II Menu

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**Which of the following is NOT a requirement of a controlled experiment?**

II.10. control comparison replication randomization all of these are required Topic II Menu

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SOLUTION Which of the following is NOT a requirement of a controlled experiment? II.10. control comparison replication randomization all of these are required Topic II Menu

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**A randomized block design is NOT:**

II.11. similar to a stratified random sample for surveys a strategy to control for an influence that would affect the outcome of the experiment a strategy that depends on randomization only used for gender comparisons all of these describe a randomized block design. Topic II Menu

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**A randomized block design is NOT:**

SOLUTION A randomized block design is NOT: II.11. You can block with any categorical variables! similar to a stratified random sample for surveys a strategy to control for an influence that would affect the outcome of the experiment a strategy that depends on randomization only used for gender comparisons all of these describe a randomized block design. Topic II Menu

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A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. The population of interest is: II.12. the 500 students chosen the students taught by activity-based statistics the students taught by traditional methods all students in high school. none of these Topic II Menu

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**We are interested in knowing about ALL AP STATISTICS STUDENTS**

SOLUTION A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. The population of interest is: II.12. We are interested in knowing about ALL AP STATISTICS STUDENTS the 500 students chosen the students taught by activity-based statistics the students taught by traditional methods all students in high school. none of these Topic II Menu

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A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. An appropriate design for the study is: II.13. a blocked design experiment a stratified random sample a completely randomized design a simple random sample none of these Topic II Menu

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SOLUTION A research team is comparing performance in AP Statistics based on whether traditional or activity-based instruction methods were used. The final grades of 500 students will be collected. An appropriate design for the study is: II.13. B or C. Either one is fine. a blocked design experiment a stratified random sample a completely randomized design a simple random sample none of these Topic II Menu

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II.14. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of freshmen, sophomores, juniors, and seniors in the sample. Topic II Menu

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**Select a simple random sample of 10 from each class.**

SOLUTION II.14. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of freshmen, sophomores, juniors, and seniors in the sample. Select a simple random sample of 10 from each class. Topic II Menu

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II.15. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of males and females in the sample Topic II Menu

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**Select a simple random sample of 20 males and 20 females.**

SOLUTION II.15. A survey is to be conducted in your school. There is to be a total of 40 students in the sample. Describe how you would choose the participants if there are to be the same number of males and females in the sample Select a simple random sample of 20 males and 20 females. Topic II Menu

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**TOPIC III: Probability**

START TOPIC Random Variables Binomial Distributions Geometric Distributions Sampling Distributions Topic III 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 back to main

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If 3 people, Joe, Betsy, and Sue, play a game in which Joe has a 25% chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win? III.1. 25% 35% 40% 65% cannot be determined Topic III Menu

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**Assuming one winner, the probabilities must add up to 100%**

SOLUTION III.1. If 3 people, Joe, Betsy, and Sue, play a game in which Joe has a 25% chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win? 25% 35% 40% 65% cannot be determined Assuming one winner, the probabilities must add up to 100% Topic III Menu

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III.2. A local law enforcement agency published the following chart. The percentage of altercations involving at least one teenager is: Altercations Between Percent Two teens 45% A teen and an adult 37% Two adults 18% 8% 37% 45% 55% 82% Topic III Menu

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**p(two teens) + p(teen and adult) =**

SOLUTION III.2. A local law enforcement agency published the following chart. The percentage of altercations involving at least one teenager is: p(two teens) + p(teen and adult) = Altercations Between Percent Two teens 45% A teen and an adult 37% Two adults 18% 8% 37% 45% 55% 82% Topic III Menu

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**What proportion of Republicans are over 50?**

Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.3. What proportion of Republicans are over 50? 61/238 32/96 96/238 32/61 cannot be determined Topic III Menu

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**What proportion of Republicans are over 50?**

SOLUTION Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.3. What proportion of Republicans are over 50? 61/238 32/96 96/238 32/61 cannot be determined There are 96 Republicans of whom 32 are over age 50 Topic III Menu

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Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.4. If one person is chosen at random, what is the probability he is a Democrat between 41 and 50 years old? 17/238 17/88 61/238 17/61 88/238 Topic III Menu

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**Total # of adults is 238. 17 are democrats between 41 and 50**

SOLUTION Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.4. If one person is chosen at random, what is the probability he is a Democrat between 41 and 50 years old? Total # of adults is are democrats between 41 and 50 17/238 17/88 61/238 17/61 88/238 Topic III Menu

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Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.5. Given that a person chosen is between 31 and 40, what is the probability the person is an Independent? 10/238 10/63 10/54 54/238 63/238 Topic III Menu

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**There are 63 people between 31 and 40 of whom 10 are Independent.**

SOLUTION Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.5. Given that a person chosen is between 31 and 40, what is the probability the person is an Independent? 10/238 10/63 10/54 54/238 63/238 There are 63 people between 31 and 40 of whom 10 are Independent. Topic III Menu

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**What proportion of the citizens sampled are over 50 OR Independent?**

Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.6. What proportion of the citizens sampled are over 50 OR Independent? 54/238 61/238 100/238 115/238 cannot be determined Topic III Menu

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**What proportion of the citizens sampled are over 50 OR Independent?**

SOLUTION Democrat Republican Independ 18 – 30 25 18 12 31 – 40 32 21 10 41 – 50 17 over 50 14 15 III.6. What proportion of the citizens sampled are over 50 OR Independent? 61/ /238 – 15/238 15 adults are in both categories 54/238 61/238 100/238 115/238 cannot be determined Topic III Menu

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**If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08 Which is true?**

III.7. Events A and B are independent and mutually exclusive Events A and B are independent but not mutually exclusive Events A and B are mutually exclusive but not independent Events A and B are neither independent nor mutually exclusive Events A and B are independent but whether they are mutually exclusive cannot be determined. Topic III Menu

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**If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08 Which is true?**

SOLUTION If P(A)=0.4, P(B)=0.2, and P(A and B)= 0.08 Which is true? III.7. Events A and B are independent and mutually exclusive Events A and B are independent but not mutually exclusive Events A and B are mutually exclusive but not independent Events A and B are neither independent nor mutually exclusive Events A and B are independent but whether they are mutually exclusive cannot be determined. --Since P(A and B) is non-zero, they are not mutually exclusive. --Since P(A and B) does not equal P(A)P(B), the events are independent—they just happen in sequence Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

What is the probability that a family with 6 children will have 3 boys and 3 girls? III.8. POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

SOLUTION What is the probability that a family with 6 children will have 3 boys and 3 girls? III.8. This is a binomial distribution: n=6 p=0.5 6C3 (0.5)3 (0.5)3 = x=3 POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

What is the probability that a person is over 6 feet tall if the mean height of her age group is 5’6” and a standard deviation of 10”? III.9. POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

SOLUTION What is the probability that a person is over 6 feet tall if the mean height of her age group is 5’6” and a standard deviation of 10”? III.9. This is a normal distribution: normalcdf(6, 1E99, 5+6/12, 10/12) = POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

III.10. What is the probability that a shipment of 100 fruit will have no more than 6 rotten fruits if the probability that any one fruit is rotten is 0.04? POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

SOLUTION III.10. What is the probability that a shipment of 100 fruit will have no more than 6 rotten fruits if the probability that any one fruit is rotten is 0.04? This is a binomial distribution: binomcdf(100, 0.04, 6) = Did you think to use the normal approximation? μ = np = 4 σ=√p(1–p)/n = .0196 normalcdf(-1E99, 6, 4, .0196) = 1.00 How come that didn’t work? POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

What is the probability that the first base hit will occur during the fourth at-bat if the probability that the hitter gets a base hit is 0.27 for any at-bat? III.11. POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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**POSSIBLE NAME OF DISTRIBUTION?**

SOLUTION What is the probability that the first base hit will occur during the fourth at-bat if the probability that the hitter gets a base hit is 0.27 for any at-bat? III.11. This is a geometric (waiting time) distribution: n=??? (there isn’t one! That’s why it’s Geometric!) p=0.27; x=4 geometpdf(0.27, 4) = (.73)(.73)(.73)(.27) = POSSIBLE NAME OF DISTRIBUTION? Binomial Model Geometric Model Uniform Model Normal Model Topic III Menu

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III.12. Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial. Find the probability she scores on 3 of her next 5 attempts Topic III Menu

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**This is a binomial distribution: n=5 p=0.70 5C3 (0.7)3 (0.3)2 = 0.3087 **

SOLUTION III.12. This is a binomial distribution: n=5 p=0.70 5C3 (0.7)3 (0.3)2 = x=3 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial Find the probability she scores on 3 of her next 5 attempts Topic III Menu

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III.13. Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial Find the probability that the first time she scores is on her 3rd attempt Topic III Menu

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**This is a geometric distribution: **

SOLUTION III.13. This is a geometric distribution: n=??? (we wait ‘til a success—there’s no n!) p=0.70; x=3 geometpdf(0.7, 3) = (.3)(.3)(.7) = 0.063 Suppose a basketball player scores 70% of her free throws. Assume each shot is independent and the probability is the same on each trial Find the probability that the first time she scores is on her 3rd attempt Topic III Menu

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**The Central Limit Theorem for sample means is critical because…**

III.14. It states that for large sample sizes, the population distribution is approximately normal It states that for large sample sizes, the sample is approximately normal It states that for any population, the sampling distribution is normal regardless of sample size It states that for large sample sizes, the sampling distribution is approximately normal regardless of the population distribution It states that for any sample size, the sampling distribution is normal Topic III Menu

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**The Central Limit Theorem for sample means is critical because…**

SOLUTION The Central Limit Theorem for sample means is critical because… III.14. It states that for large sample sizes, the population distribution is approximately normal It states that for large sample sizes, the sample is approximately normal It states that for any population, the sampling distribution is normal regardless of sample size It states that for large sample sizes, the sampling distribution is approximately normal regardless of the population distribution It states that for any sample size, the sampling distribution is normal Topic III Menu

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III.15. The amount of time it takes a high school class of 1000 freshmen to swim 10 lengths of the school pool has a distribution that is skewed left due to some excellent swimmers. The mean amount of time needed is 9.2 minutes and the standard deviation is 5.3 min. If 64 students are chosen at random, then what is the probability their mean time will exceed 10 minutes? Topic III Menu

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SOLUTION III.15. The amount of time it takes a high school class of 1000 freshmen to swim 10 lengths of the school pool has a distribution that is skewed left due to some excellent swimmers. The mean amount of time needed is 9.2 minutes and the standard deviation is 5.3 min. If 64 students are chosen at random, then what is the probability their mean time will exceed 10 minutes? normalcdf(10, 1E99, 9.2, 5.3/√64) = 0.113 Topic III Menu

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III.16. It is assumed that 50% of all people catch one or more colds each year. What is the probability that out of 400 randomly selected people, 216 or more will catch one or more colds this year? 0.0055 0.0121 0.055 0.11 0.55 Topic III Menu

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SOLUTION III.16. It is assumed that 50% of all people catch one or more colds each year. What is the probability that out of 400 randomly selected people, 216 or more will catch one or more colds this year? We want p(p > 216/400 = .054) Since np and n(1-p) exceed 10, we can apply the normal approximation with µ=.5 and σ=.025 Normalcdf(.54, 1E99, .5, .025) = .055 0.0055 0.0121 0.055 0.11 0.55 Topic III Menu

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III.17. The decision whether the distribution ofo a sample mean follows a normal or a t-distribution depends on: Sample size Whether you have the actual data or only statistics of the data Whether you know the population standard deviation Whether np>10 and n(1-p)>10 None of the above Topic III Menu

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SOLUTION III.17. The decision whether the distribution ofo a sample mean follows a normal or a t-distribution depends on: Sample size Whether you have the actual data or only statistics of the data Whether you know the population standard deviation Whether np>10 and n(1-p)>10 None of the above Topic III Menu

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National studies show that 14% of male teenagers and 12% of female teenagers will be involved in a major traffic accident while driving. What’s the probability that independent samples of 100 female teens and 75 male teens will have results that differ by more than 3% in either direction? III.18. .042 .085 .42 .85 Cannot be determined Topic III Menu

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**For both genders, np and n(1-p) exceed 5 so we can use the CLT.**

SOLUTION National studies show that 14% of male teenagers and 12% of female teenagers will be involved in a major traffic accident while driving. What’s the probability that independent samples of 100 female teens and 75 male teens will have results that differ by more than 3% in either direction? III.18. For both genders, np and n(1-p) exceed 5 so we can use the CLT. z = ( ) = .1938 √(.14*.86/ *.88/100) Using normalcdf(.1938, 1E99) = .423 But we want both directions so add in the other tail to get an answer of .846 .042 .085 .42 .85 Cannot be determined

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**TOPIC IV: Inference START TOPIC Inference on Means**

Inference on Proportions Inference on 2-way Tables Inference on Regression START TOPIC Topic IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 back to main

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**Given H0: μ=30, HA: μ<30, if you conclude that the mean is less than 30 when it is actually 27…**

you have made a type II error you have made a type I error the result of your test was not significant you have drawn a correct conclusion all of the above are true Topic IV Menu

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**You are rejecting a false null hypothesis. No problems. **

SOLUTION Given H0: μ=30, HA: μ<30, if you conclude that the mean is less than 30 when it is actually 27… IV.1. you have made a type II error you have made a type I error the result of your test was not significant you have drawn a correct conclusion all of the above are true You are rejecting a false null hypothesis. No problems. Correct decision. Topic IV Menu

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The local news station reports that the 97% confidence interval for a candidate’s support was (43%, 48%). What does the phrase “97% confidence” mean? IV.2. 97% of the voters support the candidate 97% of the time, this candidate’s level of support will be between 43% and 48% There is a 97% probability that the true level of support is between 43% and 48% There is a 97% probability that any other sample percentage is in the interval (43%, 48%) none of these is true Topic IV Menu

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**Correct interpretation:**

SOLUTION The local news station reports that the 97% confidence interval for a candidate’s support was (43%, 48%). What does the phrase “97% confidence” mean? IV.2. not (D) because you’re predicting the range of the TRUE proportion; you’re not interested in predicting the range of other samples!!! Correct interpretation: “If this process were repeated over and over again, about 97% of our INTERVALS would capture the true proportion.” 97% of the voters support the candidate 97% of the time, this candidate’s level of support will be between 43% and 48% There is a 97% probability that the true level of support is between 43% and 48% There is a 97% probability that any other sample percentage is in the interval (43%, 48%) none of these is true Topic IV Menu

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A study of 20 teachers in a school district indicated that the 95% confidence interval for the mean salary of all teachers is ($38,945, $41,245). What assumptions must be true for this interval to be valid? IV.3. no assumptions are necessary. The CLT applies The sample is randomly selected from a population of salaries that is a t-distribution the distribution of the sample means is approximately normal the distribution of all teachers’ salaries is approximately normal None of the above. Topic IV Menu

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SOLUTION A study of 20 teachers in a school district indicated that the 95% confidence interval for the mean salary of all teachers is ($38,945, $41,245). What assumptions must be true for this interval to be valid? IV.3. Because our sample size is only 20, you have to know the population was normal to begin with. (D) is the best response. …I might accept choice (C) too since the CLT is all about when x-bar is normal, which is the goal I suppose. no assumptions are necessary. The CLT applies The sample is randomly selected from a population of salaries that is a t-distribution the distribution of the sample means is approximately normal the distribution of all teachers’ salaries is approximately normal None of the above. Topic IV Menu

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**In order to reduce the width of a confidence interval, we can:**

IV. 4. increase sample size only increase confidence level only increase sample size and increase confidence level increase sample size and decrease confidence level none of these would reduce the width of the interval Topic IV Menu

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**In order to reduce the width of a confidence interval, we can:**

SOLUTION In order to reduce the width of a confidence interval, we can: IV. 4. Less confidence affords you to predict a much tighter range of values. increase sample size only increase confidence level only increase sample size and increase confidence level increase sample size and decrease confidence level none of these would reduce the width of the interval Topic IV Menu

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If the 95% confidence interval for μ is (6,9), what conclusion can we draw if we test Ho: μ=10 vs. Ha: μ≠10 at α=.05? IV.5. reject Ho fail to reject Ho accept Ho accept Ha There is insufficient information given to draw a conclusion Topic IV Menu

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SOLUTION If the 95% confidence interval for μ is (6,9), what conclusion can we draw if we test Ho: μ=10 vs. Ha: μ≠10 at α=.05? IV.5. A two tailed test of α=.05 is equivalent to a 95% confidence interval. 10 is not within the interval. reject Ho fail to reject Ho accept Ho accept Ha There is insufficient information given to draw a conclusion Topic IV Menu

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A bakery determines that it will be profitable if the time it takes to decorate a cake does not exceed 45 minutes. The owner documents the time spent on 20 cakes and performs a test. If the P-value of the test is 0.032, then he should conclude: IV.6. at α=.05, fail to reject Ho at α=.05, reject Ho at α=.03, reject Ho at α=.025, reject Ho We cannot draw a conclusion from this information Topic IV Menu

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**a p-value of .032 is significant at the 5% level**

SOLUTION A bakery determines that it will be profitable if the time it takes to decorate a cake does not exceed 45 minutes. The owner documents the time spent on 20 cakes and performs a test. If the P-value of the test is 0.032, then he should conclude: IV.6. at α=.05, fail to reject Ho at α=.05, reject Ho at α=.03, reject Ho at α=.025, reject Ho We cannot draw a conclusion from this information a p-value of .032 is significant at the 5% level Topic IV Menu

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IV.7. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What design seems to be employed? boys girls n 110 135 Mean 71.6 68.3 St.Deviation 10.4 11.2 Matched Pairs Design Simple Random Design Multi-State Cluster Design Independent Samples Design Randomized Block Design Topic IV Menu

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**We have 2 independent samples here: boys and girls.**

SOLUTION IV.7. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What design seems to be employed? We have 2 independent samples here: boys and girls. boys girls n 110 135 Mean 71.6 68.3 St.Deviation 10.4 11.2 Matched Pairs Design Simple Random Design Multi-State Cluster Design Independent Samples Design Randomized Block Design Topic IV Menu

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IV.8. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What are Ho and Ha to determine if boys’ scores are higher than girls? boys girls n 110 135 Mean 71.6 68.3 St.Dev 10.4 11.2 Ho: μb – μg = 0; Ha: μb – μg < 0 Ho: μb – μg = 0; Ha: μb – μg ≠ 0 Ho: μb – μg = 0; Ha: μb – μg > 0 Ho: μb – μg < 0; Ha: μb – μg = 0 Ho: μb = μg; Ha: μb ≠ μg Topic IV Menu

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**Another way to write this would be Ho: μb = μg; Ha: μb > μg**

SOLUTION IV.8. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. What are Ho and Ha to determine if boys’ scores are higher than girls? Another way to write this would be Ho: μb = μg; Ha: μb > μg boys girls n 110 135 Mean 71.6 68.3 St.Dev 10.4 11.2 Ho: μb – μg = 0; Ha: μb – μg < 0 Ho: μb – μg = 0; Ha: μb – μg ≠ 0 Ho: μb – μg = 0; Ha: μb – μg > 0 Ho: μb – μg < 0; Ha: μb – μg = 0 Ho: μb = μg; Ha: μb ≠ μg Topic IV Menu

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IV.9. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. Suppose the p-value is We can then conclude: at α=.025, reject Ho at α=.02, reject Ho at α=.01, reject Ho at α=.025, fail to reject Ho We cannot draw a conclusion from this information boys girls n 110 135 Mean 71.6 68.3 St.Dev 10.4 11.2 Topic IV Menu

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SOLUTION IV.9. Do boys perform better in math than girls? A randomly selected group of each gender were given the same math assessment. Suppose the p-value is We can then conclude: .0344 is not significant at .025, .02, or We must fail to reject it at these levels. at α=.025, reject Ho at α=.02, reject Ho at α=.01, reject Ho at α=.025, fail to reject Ho We cannot draw a conclusion from this information boys girls n 110 135 Mean 71.6 68.3 St.Dev 10.4 11.2 Topic IV Menu

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IV.10. The rejection region for a test Ho: p=.4 vs. Ha: p<.4, with n=50 and α=.05 is given by: Reject Ho if z > 1.96 or z < -1.96 Reject Ho if z > or z < Reject Ho if z < -1.96 Reject Ho if z < Reject Ho if t > 2.59 Topic IV Menu

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**because the entire .05 rejection region is in the left tail.**

SOLUTION IV.10. The rejection region for a test Ho: p=.4 vs. Ha: p<.4, with n=50 and α=.05 is given by: invnorm(.05) because the entire .05 rejection region is in the left tail. Reject Ho if z > 1.96 or z < -1.96 Reject Ho if z > or z < Reject Ho if z < -1.96 Reject Ho if z < Reject Ho if t > 2.59 Topic IV Menu

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**The power of a test is described by all of the following EXCEPT**

IV.11. Power = p( rejecting Ho when Ha is true) Power = 1 – β Power = α + β The calculation of power requires knowing the values of μ0, μa, σ, and α All of these are correct descriptions of the concept of power. Topic IV Menu

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**The power of a test is described by all of the following EXCEPT**

SOLUTION The power of a test is described by all of the following EXCEPT IV.11. But it IS true that as one increases, the other has to decrease. …just not always by the same amount Power = p( rejecting Ho when Ha is true) Power = 1 – β Power = α + β The calculation of power requires knowing the values of μ0, μa, σ, and α All of these are correct descriptions of the concept of power. Topic IV Menu

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**The population variances should be equal **

In a matched pairs test of 75 pairs, which of the following assumptions is necessary? IV.12. The distribution of the paired differences should be approximately normal The population variances should be equal The samples are randomly and independently selected The sets of values for each variable are approximately normal None of these assumptions is necessary. Topic IV Menu

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**The population variances should be equal **

SOLUTION In a matched pairs test of 75 pairs, which of the following assumptions is necessary? IV.12. The distribution of the paired differences should be approximately normal The population variances should be equal The samples are randomly and independently selected The sets of values for each variable are approximately normal None of these assumptions is necessary. And being good stats students, you will either make sure n>25 or do a quick little graph to show there aren’t any outliers. Right? Topic IV Menu

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IV.13. A pharmaceutical company claims that 50% of adult males living in a city in the Midwest get at least two colds per year. A random sample of 100 adult males living in the city reported that only 42% got two or more colds. Do these data provide evidence (at the 5% significance level) that the true proportion of people is less than 50% Topic IV Menu

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**Of course you would want to run through the entire test properly, but…**

SOLUTION IV.13. Of course you would want to run through the entire test properly, but… we fail to reject at the 5% level. there is not enough evidence to say the proportion is less than 50% A pharmaceutical company claims that 50% of adult males living in a city in the Midwest get at least two colds per year. A random sample of 100 adult males living in the city reported that only 42% got two or more colds. Do these data provide evidence (at the 5% significance level) that the true proportion of people is less than 50% Topic IV Menu

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**The confidence interval is not valid **

If a 90% confidence interval for the slope of a regression line does not contain 0, then which of the following is a valid conclusion? IV.14. The confidence interval is not valid A significance test will not be significant at the 10% level There is sufficient evidence to conclude that the slope of the true regression line is 0 There is sufficient evidence to conclude that the slope of the true regression line is not 0. None of these is valid. Topic IV Menu

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**The confidence interval is not valid **

SOLUTION If a 90% confidence interval for the slope of a regression line does not contain 0, then which of the following is a valid conclusion? IV.14. If 0 isn’t in the interval, we are 90% confident that the slope is NOT zero. This is exactly what a regression test would conclude: the slope is NOT zero. The confidence interval is not valid A significance test will not be significant at the 10% level There is sufficient evidence to conclude that the slope of the true regression line is 0 There is sufficient evidence to conclude that the slope of the true regression line is not 0. None of these is valid. Topic IV Menu

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**What is the expected number of males who prefer chocolate?**

IV.15. What is the expected number of males who prefer chocolate? 27.8 29.2 31.3 36.3 None of these male female chocolate 32 16 vanilla 14 4 strawberry 3 10 Topic IV Menu

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**What is the expected number of males who prefer chocolate?**

48 of 79 people prefer chocolate (60.76%), so we expect 60.76% of the 49 males to like chocolate. .6076(49) = 29.77 or if you prefer the book formula: (row total x column total) total number = 29.77 SOLUTION IV.15. What is the expected number of males who prefer chocolate? 27.8 29.2 31.3 36.3 None of these male female chocolate 32 16 vanilla 14 4 strawberry 3 10 Topic IV Menu

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Section 10.1 Estimating with Confidence AP Statistics February 11 th, 2011.

Section 10.1 Estimating with Confidence AP Statistics February 11 th, 2011.

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