# AP * Statistics Review

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this material.

2 Page 1 of 23 A Confidence Interval is an interval that is computed from sample data and provides a range of plausible values for a population parameter. A Confidence Level is a number that provides information on how much confidence we have in the method used to construct a confidence interval estimate. This level specifies the percentage of all possible samples that produce an interval containing the true value of the population parameter. Constructing a Confidence Interval The steps listed below should be followed when asked to calculate a confidence interval. 1. Identify the population of interest and define the parameter of interest being estimated. 2. Identify the appropriate confidence interval by name or formula. 3. Verify any conditions (assumptions) that need to be met for that confidence interval. 4. Calculate the confidence interval. 5. Interpret the interval in the context of the situation. The general formula for a confidence interval calculation is: Statistic ± (critical value) (standard deviation of statistic) The critical value is determined by the confidence level. The type of statistic is determined by the problem situation. Here we will discuss two types of statistics: mean and proportion. Confidence Interval for Proportions: (1-proportion z-interval) We are finding an interval that describes the population proportion ( p or π ). The general formula uses the sample proportion ( ˆp ) and the sample size ( n ). Statistic ± (critical value) (standard deviation of statistic) pˆ(1 pˆ) pˆ ± ( z* ) n The conditions that need to be met for this procedure are: 1. The sample is a simple random sample. 2. The population is large relative to the sample. 10n< N (N = the size of the population) 3. The sampling distribution of the sample proportion is approximately normal. np 10 n(1 p) 10 Complete the calculations after showing that the conditions are met. Write the answer in the context of the original problem.

3 Page 2 of 23 Sample question: The owner of a popular chain of restaurants wishes to know if completed dishes are being delivered to the customer s table within one minute of being completed by the chef. A random sample of 75 completed dishes found that 60 were delivered within one minute of completion. Find the 95% confidence interval for the true population proportion. 1. Identify the population of interest and define the parameter of interest being estimated. The population of interest is the dishes that are being served at this chain of restaurants. p = the population proportion of dishes that are served within one minute of completion. 60 p ˆ = = 0.8 = the sample proportion of dishes that are served within one minute 75 of completion. n = 75 is the sample size. 2. Identify the appropriate confidence interval by name or formula. We will use a 95% confidence z-interval for proportions. 3. Verify any conditions (assumptions) that need to be met for that confidence interval. The problem states that this is a simple random sample. 10n< N 10(75) < N 750 < N It is reasonable to assume that a popular chain of restaurants will serve more than 750 dishes. npˆ 10 n(1 pˆ ) 10 75(0.8) = 60 75(1.8) = It is reasonable to use a normal model. 4. Calculate the confidence interval. At a 95% CI, the critical value is z * = This value is found on the last row of the t-distribution table. 0.8(1 0.8) 0.8 ± ± (0.709,0.891) 5. Interpret the interval in the context of the situation. We are 95% confident that the true proportion of dishes that are served within one minute of completion for this chain of restaurants is between and

4 Page 3 of 23 Confidence Interval for Means with σ known: (z-interval) We are finding an interval that describes the population mean ( μ ). The general formula uses the sample mean ( x ), the population standard deviation ( σ ) and the sample size ( n ). Statistic ± (critical value) (standard deviation of statistic) σ x ± ( z* ) n The conditions that need to be met for this procedure are: 1. The sample is a simple random sample. 2. The population is normal or n The population standard deviation (σ ) is known. Complete the calculations after showing that the conditions are met. Write your answer in the context of the original problem. Sample question: An asbestos removal company places great importance on the safety of their employees. The protective suits that the employees wear are designed to keep asbestos particles off the employee s body. The owner is interested in knowing the average amount of asbestos particles left on the employee s skin after a days work. A random sample of 100 employees had skin tests after removing their protective suit. The average number of particles found was.481 particles per square centimeter. Assuming that the population standard deviation is 0.35 particles per square centimeter, calculate a 95% confidence interval for the number of particles left on the employee s skin. Solution: The population of interest is the employees of the asbestos removal company. μ = the population mean of particles per square centimeter after a days work wearing the protective suit. x = = the sample mean of the number of particles per square centimeter. n = 100 is the sample size. σ = 0.35 =population standard deviation. We will use a 95% confidence z-interval for means (z-interval). The problem states that this is a simple random sample. n = 100 is greater than 30 so we can assume that use of a normal model is reasonable. At a 95% CI, the critical value is * 1.96 z =. This value is found on the last row of the t-distribution table.

5 ± ± (0.412,0.550) Confidence Intervals Page 4 of 23 We are 95% confident that the true mean number of particles of asbestos found on the skin of an employee after a days work is between and particles per square centimeter. Confidence Interval for Means with σ unknown: (t-interval) You will be finding an interval that will describe the population mean ( μ ). The general formula will use the sample mean ( x ), the sample standard deviation ( s ), the sample size ( n ), and the degrees of freedom ( n 1 ). Statistic ± (critical value) (standard deviation of statistic) s x ± ( t* ) n The conditions that need to be met for this procedure are: 1. The sample is a simple random sample. 2. The population is approximately normal (graphical support required) or n The population standard deviation (σ ) is unknown. Complete the calculations after showing that the conditions are met. Write your answer in the context of the original problem. This situation (where the population standard deviation is unknown) is much more realistic that the previous case. Sample question: A biology student at a major university is writing a report about bird watchers. She has developed a test that will score the abilities of a bird watcher to identify common birds. She collects data from a random sample of people that classify themselves as bird watchers (data shown below). Find a 90% confidence interval for the mean score of the population of bird watchers

6 Page 5 of 23 Solution: The population of interest is people that classify themselves as bird watchers. μ = the population mean score on the bird identification ability test. x = = the sample mean of the scores on the ability test. s = = sample standard deviation of test scores. n = 20 is the sample size. df = 20 1 = 19 degrees of freedom. We will use a 90% confidence t-interval for means (t-interval). The problem states that this is a simple random sample. The sample size, 20, is smaller than 40 so we will assess normality by looking at the normal probability plot. The normal probability plot appears to be linear so we will assume an approximately normal distribution of scores. The population standard deviation is unknown. At a 90% CI, the critical value is t * = This value is found on the t-distribution table using 19 degrees of freedom ± ± (6.289,7.281) We are 90% confident that the true mean score on the bird identification ability test of the population of persons that classify themselves as bird watchers is between and

7 Page 6 of 23 Summary of Confidence Intervals with One Sample Confidence Interval Type Proportions Formula Conditions Calculator Test ( z ) pˆ ± * pˆ(1 pˆ) n 1. The sample is a simple random sample. 2. The population is large relative to the sample 10n< N 3. np 10 n(1 p) 10 1-PropZInterval Means σ ± n (σ known) x ( z* ) Means s ± n (σ unknown) x ( t* ) 1. The sample is a simple random sample. 2. The population is normal or n The population standard deviation (σ ) is known. 1. The sample is a simple random sample. 2. The population is approximately normal (graphical support required) or n The population standard deviation (σ ) is unknown. ZInterval TInterval Confidence Intervals with TWO Samples We may be asked to work a confidence interval problem to estimate the difference of two population parameters. The process is very similar to that of a one sample confidence interval. We must make sure that the conditions are met for both samples and that the samples are independent of each other.

8 Page 7 of 23 The formulas for the two sample confidence intervals are as follows: Confidence Interval Type Difference of Proportions p p ( ) 1 2 Difference of Means (σ known) μ μ ( ) 1 2 Difference of Means (σ unknown) μ μ ( ) 1 2 Formula Conditions Calculator Test pˆ1(1 pˆ1) pˆ ˆ 2(1 p2) pˆ ˆ 1 p2 ± ( z* ) + n n 1 2 σ σ x1 x2 ± ( z* ) + n n s s x1 x2 ± ( t* ) + n n Note: The formula for degrees of freedom is 2 2 s1 s 2 + n n df = s1 s2 n1 n2 + n 1 n This formula is not provided on the formula chart. A more conservative quantity used by some statisticians is the smaller of the two individual degrees of freedom; either n1 1 or n The samples are 2-PropZInterval independently selected simple random samples. 2. Each population is large relative to the sample 10n1 < N1 10n2 < N2 3. np n1(1 p1) 10 np n2(1 p2) The samples are 2-samp-ZInterval independently selected simple random samples. 2. Both populations are normal or n1 30 n Both population standard deviations ( σ1and σ 2 ) are known. 1. The samples are 2-samp-TInterval independently selected simple random samples. 2. Both populations are approximately normal (graphical support is required) or n1 40 n Both population standard deviations ( σ1and σ 2 ) are unknown.

9 Page 8 of 23 Sample question: Two popular strategy video games, AE and C, are known for their long play times. A popular game review website is interested in finding the mean difference in play time between these games. The website selects a random sample of 43 gamers to play AE and finds their sample mean play time to be 3.6 hours with a standard deviation of 0.9 hours. The website also selected a random sample of gamers to test the game C. There test included 40 gamers with a sample mean of 3.1 hours and a standard deviation of 0.4 hours. Find the 98% confidence interval for the difference μ AE μ C. Solution: The population of interest is play time of games AE and C. μ AE = the population mean play time for game AE. μ C = the population mean play time for game C. x = 3.6 = the sample mean of play time for game AE. AE x C = 3.1 = the sample mean of play time for game C. s AE = 0.9 = sample standard deviation play time for game AE. s C = 0.4 = sample standard deviation play time for game C. n AE = 43 is the sample size of the players of game AE. n C = 40 is the sample size of the players of game C. df = 43 1 = 42 degrees of freedom (using the simplified approximation for df). We will use a 98% confidence t-interval for the difference of means (2-sample-tinterval). The problem states that each is a simple random sample. The sample sizes are 43 and 40 which are both at least 40. The population standard deviation is unknown. At a 98% CI, the critical value is t * = This value is found on the t-distribution table using 40 degrees of freedom (because the table does not have a value for 42) ( ) ± ± (0.1338,0.8662) 2 2 We are 98% confident that the true difference in mean play time between games AE and C falls between and hours.

10 Page 9 of 23 Note: The calculator gives a result of (0.1386, ) for this confidence interval. The difference in these answers is due to its use of degrees of freedom. Our answer has a slightly wider spread, and thus is more conservative. Other Confidence Interval Topics: Statistic ± (critical value) (standard deviation of statistic) Margin of Error (ME) the margin of error is the value of the critical value times the standard deviation of the statistic. It is the plus or minus part of the confidence interval. Some problems might ask you to determine the sample size required given a margin of error. This requires a little algebra to work backward through the equations. The equations are listed below. Confidence Interval Type Formula for finding the sample size within a margin of error ME 1-sample proportion 2 z * n= pˆ(1 pˆ) ME Note: if ˆp is not given in this type of problem, a conservative value to use if sample mean (z-interval) 2 z * σ n = ME Keep in mind the effects of changing the confidence level. A large confidence level (say 99% as compared to 90%) produces a larger margin of error. To be more confident we must include more values in our range. Do not confuse the meaning of a confidence level: A 95% confidence level means that if we repeated the sampling process many times, the resulting confidence interval would capture the true population parameter 95% of the time.

11 Page 10 of 23 Multiple Choice Questions on Confidence Intervals 1. A random sample of 100 visitors to a popular theme park spent an average of \$142 on the trip with a standard deviation of \$47.5. Which of the following would the 98% confidence interval for the mean money spent by all visitors to this theme park? (A) (\$130.77, \$153.23) (B) (\$132.57, \$151.43) (C) (\$132.69, \$151.31) (D) (\$140.88, \$143.12) (E) (\$95.45, \$188.55) 2. How large of a random sample is required to insure that the margin of error is 0.08 when estimating the proportion of college professors that read science fiction novels with 95% confidence? (A) 600 (B) 300 (C) 150 (D) 75 (E) A quality control specialist at a plate glass factory must estimate the mean clarity rating of a new batch of glass sheets being produced using a sample of 18 sheets of glass. The actual distribution of this batch is unknown, but preliminary investigations show that a normal approximation is reasonable. The specialist decides to use a t- distribution rather than a z-distribution because (A) The z-distribution is not appropriate because the sample size is too small. (B) The sample size is large compared to the population size. (C) The data comes from only one batch. (D) The variability of the batch is unknown. (E) The t-distribution results in a narrower confidence interval.

13 Page 12 of A research and development engineer is preparing a report for the board of directors on the battery life of a new cell phone they have produced. At a 95% confidence level, he has found that the battery life is 3.2 ± 1.0 days. He wants to adjust his findings so the margin of error is as small as possible. Which of the following will produce the smallest margin of error? (A) Increase the confidence level to 100%. This will assure that there is no margin of error. (B) Increase the confidence level to 99%. (C) Decrease the confidence level to 90%. (D) Take a new sample from the population using the exact same sample size. (E) Take a new sample from the population using a smaller sample size. 7. A biologist has taken a random sample of a specific type of fish from a large lake. A 95 percent confidence interval was calculated to be 6.8 ± 1.2 pounds. Which of the following is true? (A) 95 percent of all the fish in the lake weigh between 5.6 and 8 pounds. (B) In repeated sampling, 95 percent of the sample proportions will fall within 5.6 and 8 pounds. (C) In repeated sampling, 95% of the time the true population mean of fish weights will be equal to 6.8 pounds. (D) In repeated sampling, 95% of the time the true population mean of fish weight will be captured in the constructed interval. (E) We are 95 percent confident that all the fish weigh less than 8 pounds in this lake. 8. A polling company is trying to estimate the percentage of adults that consider themselves happy. A confidence interval based on a sample size of 360 has a larger than desired margin of error. The company wants to conduct another poll and obtain another confidence interval of the same level but reduce the error to one-third the size of the original sample. How many adults should they now interview? (A) 40 (B) 180 (C) 720 (D) 1080 (E) 3240

14 Page 13 of A researcher is interested in determining the mean energy consumption of a new compact florescent light bulb. She takes a random sample of 41 bulbs and determines that the mean consumption is 1.3 watts per hour with a standard deviation of 0.7. When constructing a 97% confidence interval, which would be the most appropriate value of the critical value? (A) (B) (C) (D) (E) A 98 percent confidence interval for the mean of a large population is found to be 978 ± 25. Which of the following is true? (A) 98 percent of all observations in the population fall between 953 and 1003 (B) The probability of randomly selecting an observation between 953 and 1003 from the population is 0.98 (C) If the true population mean is 950, then this sample mean of 978 would be unlikely to occur. (D) If the true population mean is 990, then this sample mean of 978 would be unexpected. (E) If the true population mean is 1006, then this confidence interval must have been calculated incorrectly.

15 Page 14 of 23 Free Response Questions on Confidence Intervals Free Response 1. A random sample of 9 th grade math students was asked if they prefer working their math problems using a pencil or a pen. Of the 250 students surveyed, 100 preferred pencil and 150 preferred pen. (a) Using the results of this survey, construct a 95 percent confidence interval for the proportion of 9 th grade students that prefer to work their math problems in pen. (b) A school newspaper reported on the results of this survey by saying, Over half of ninth-grade math students prefer to use pen on their math assignments. Is this statement supported by your confidence interval? Explain.

16 Page 15 of 23 Free Response 2. A major city in the United States has a large number of hotels. During peak travel times throughout the year, these hotels use a higher price for their rooms. A travel agent is interested in finding the difference of the average cost of a hotel rooms from the peak season to the off season. He takes a random sample of hotel room costs during each of these seasons. Plots of both samples of data indicate that the assumption of normality is not unreasonable. Season Cost Standard Deviation Sample Size Peak Season \$245 \$45 33 Off Season \$135 \$65 38 (a) Construct a 95 percent confidence interval for the difference of the mean cost of hotel rooms from peak season to the off season. (b) One particular hotel has an off season rate of \$88 and a peak season rate of \$218. Based on your confidence interval, comment on the price difference of this hotel.

17 Page 16 of 23 Key to Confidence Interval Multiple Choice 1. A Use the t-interval 2. C Make a conservative assumption that the proportion is D The standard deviation of the population (batch) is not known. 4. D 2-prop-z-interval 5. C Conditions of 1-proportion-z-interval 6. C Decreasing the confidence level decreases the margin of error 7. D Correct interpretation of the confidence level 8. E Increasing the sample size by 9 decreases the error by one-third 9. C Estimates from the t-table or invnorm(.985) produce this result. 10. C Reasonable alternative interpretation of a confidence interval

18 Page 17 of 23 Rubric for Confidence Interval Free Response Free Response 1. Solution Part (a) 1. Identify the population of interest and define the parameter of interest being estimated. The population of interest is all ninth grade math students. p = the population proportion of ninth grade math students that prefer to use pen on their math work. 150 p ˆ = = 0.6 = the sample proportion of ninth grade math students that prefer 250 to use pen. n = 250 is the sample size. 2. Identify the appropriate confidence interval by name or formula. We will use a 95% confidence z-interval for proportions. 3. Verify any conditions (assumptions) that need to be met for that inference procedure. The problem states that this is a simple random sample. 10n< N 10(250) < N 2500 < N It is reasonable to assume that there are more than 2500 ninth grade math students. npˆ 10 n(1 pˆ ) (0.6) = (1.6) = It is reasonable to use the normal approximation. 4. Calculate the confidence interval. At a 95% CI, the critical value is z * = (1 0.6) 0.6 ± ± (0.5393,0.6607)

19 Page 18 of Interpret your results in the context of the situation. We are 95% confident that the true proportion of ninth grade math students that prefer to do their math work with pen is between and Part (b) The school paper states that more than half of ninth grade math students prefer to do their math in pen. 50% or 0.5 is below our confidence interval. Because our confidence interval contains values that are above 0.5, we believe that the newspaper has made an accurate statement. Scoring Parts (a) and (b) are essentially correct (E), partially correct (P), or incorrect (I). Part (a) is correct if each of the following is included: Identify the appropriate confidence interval by name or formula Check appropriate conditions Correct mechanics Interpret the confidence interval in context Part (a) is partially correct if only three of the above are provided in the answer. Part (a) is incorrect if less than three of components are provided in the answer. Part (b) is essentially correct if the student connects the paper s claim of 0.5 to the confidence interval and says that the interval supports the paper s claim. Part (b) is partially correct if the student says that the confidence interval does support the paper s claim, but have weak or incomplete explanation. Part (b) is incorrect if the student fails to provide an explanation. 4 Complete Response Both parts essentially correct. 3 Substantial Response One part essentially correct and one part partially correct 2 Developing Response

20 OR Confidence Intervals Page 19 of 23 One part essentially correct and one part incorrect Partially correct on both parts 1 Minimal Response Partially correct on one part.

21 Free Response 2. Solution Confidence Intervals Page 20 of 23 Part (a) 1. Identify the population of interest and define the parameters of interest being estimated. The population of interest is price of hotel rooms in a major city. μ = the population mean cost of a hotel room in peak season. μ off peak = the population mean cost of a hotel room in off season. x = 245 = the sample mean cost of a hotel room in peak season. peak x off = 135 = the sample mean cost of a hotel room in off season. s peak = 45 = sample standard deviation cost of a hotel room in peak season. s off = 65 = sample standard deviation cost of a hotel room in off season. n peak = 33 is the sample size of the cost of a hotel room in peak season. n off = 38is the sample size of the cost of a hotel room in off season. df = 33 1 = 32 degrees of freedom (using the simplified approximation for df). 2. Identify the appropriate confidence interval by name or formula. We will use a 95% confidence t-interval for the difference of means (2-sample-tinterval). 3. Verify any conditions (assumptions) that need to be met for that inference procedure. The problem states that each is a simple random sample. The problem states that the assumption of normality is reasonable. The population standard deviation is unknown. 4. Calculate the confidence interval. At a 95% CI, the critical value is t * = from the table using df = 30. This value is found on the t-distribution table using 30 degrees of freedom (because the table does not have a value for 32) ± ( ) 110 ± (83.18,136.82)

22 Page 21 of Interpret your results in the context of the situation. We are 95% confident that the true difference in mean cost of a hotel room in this city between peak season and off season falls between \$83.18 and \$ Note: The calculator gives a result of (\$83.77, \$136.23) for this confidence interval. df = 65.9 Part (b) The difference in cost from peak season to off season is = \$130 Because \$130 falls inside our confidence interval, we would not think that this is an extreme change in price from peak to off season. Scoring Parts (a) and (b) are essentially correct (E), partially correct (P), or incorrect (I). Part (a) is correct if each of the following is included: Identify the appropriate confidence interval by name or formula Check appropriate conditions Correct mechanics Interpret the confidence interval in context Part (a) is partially correct if only three of the above are provided in the answer. Part (a) is incorrect if less than three of components are provided in the answer. Part (b) is essentially correct if the student compares the difference in prices to the confidence interval and concludes that this is not an unusual price change for this city. Part (b) is partially correct if the student says that this in not an unusual price change but does not directly tie their conclusion to the confidence interval. Part (b) is incorrect if fails to provide an explanation. 4 Complete Response Both parts essentially correct. 3 Substantial Response One part essentially correct and one part partially correct 2 Developing Response

23 Page 22 of 23 OR One part essentially correct and one part incorrect Partially correct on both parts 1 Minimal Response Partially correct on one part.

24 Page 23 of 23 AP Statistics Exam Connections The list below identifies free response questions that have been previously asked on the topic of Confidence Intervals. These questions are available from the CollegeBoard and can be downloaded free of charge from AP Central Free Response Questions 2002 B Question Question Question Question 4

### AP STATISTICS 2011 SCORING GUIDELINES (Form B)

AP STATISTICS 2011 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) identify and check appropriate conditions for inference;

### Chapter 7. Estimates and Sample Size

Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average

### AP STATISTICS 2011 SCORING GUIDELINES

AP STATISTICS 2011 SCORING GUIDELINES Question 6 Intent of Question The primary goals of this question were to assess students ability to (1) construct and interpret a confidence interval for a population

### Section 7.2 Confidence Intervals for Population Proportions

Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct

### Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

Sampling (cont d) and Confidence Intervals 11.220 Lecture 9 8 March 2006 R. Ryznar Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form

### AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

### AP Statistics 2013 Scoring Guidelines

AP Statistics 2013 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### Outline. 1 Confidence Intervals for Proportions. 2 Sample Sizes for Proportions. 3 Student s t-distribution. 4 Confidence Intervals without σ

Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for

### find confidence interval for a population mean when the population standard deviation is KNOWN Understand the new distribution the t-distribution

Section 8.3 1 Estimating a Population Mean Topics find confidence interval for a population mean when the population standard deviation is KNOWN find confidence interval for a population mean when the

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Confidence Intervals about a Population Mean

Confidence Intervals about a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Motivation Goal: to estimate a population mean µ based on data collected

### AP STATISTICS (Warm-Up Exercises)

AP STATISTICS (Warm-Up Exercises) 1. Describe the distribution of ages in a city: 2. Graph a box plot on your calculator for the following test scores: {90, 80, 96, 54, 80, 95, 100, 75, 87, 62, 65, 85,

### Math 140: Introductory Statistics Instructor: Julio C. Herrera Exam 3 January 30, 2015

Name: Exam Score: Instructions: This exam covers the material from chapter 7 through 9. Please read each question carefully before you attempt to solve it. Remember that you have to show all of your work

### AP Statistics Solutions to Packet 9

AP Statistics Solutions to Packet 9 Sampling Distributions Sampling Distributions Sample Proportions Sample Means HW #7 1 4, 9, 10 11 For each boldface number in Exercises 9.1 9.4, (a) state whether it

### Sociology 6Z03 Topic 15: Statistical Inference for Means

Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall 2016 1 / 41 Outline: Statistical

### Chapter 8 Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter problem: Does the MicroSort method of gender selection increase the likelihood that a baby will be girl? MicroSort: a gender-selection method developed by Genetics

### AP Statistics 2000 Scoring Guidelines

AP Statistics 000 Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought

### AP Statistics 2004 Scoring Guidelines

AP Statistics 2004 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought

### MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### AP STATISTICS 2012 SCORING GUIDELINES

2012 SCORING GUIDELINES Question 4 Intent of Question The primary goal of this question was to assess students ability to identify, set up, perform, and interpret the results of an appropriate hypothesis

### = 2.0702 N(280, 2.0702)

Name Test 10 Confidence Intervals Homework (Chpt 10.1, 11.1, 12.1) Period For 1 & 2, determine the point estimator you would use and calculate its value. 1. How many pairs of shoes, on average, do female

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

### CHAPTER 8 ESTIMATION

CHAPTER 8 ESTIMATION CONFIDENCE INTERVALS FOR A POPULATION MEAN (SECTION 8.1 AND 8.2 OF UNDERSTANDABLE STATISTICS) The TI-83 Plus and TI-84 Plus fully support confidence intervals. To access the confidence

### Chapter 7 Review. Confidence Intervals. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Chapter 7 Review Confidence Intervals MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Suppose that you wish to obtain a confidence interval for

### Estimating and Finding Confidence Intervals

. Activity 7 Estimating and Finding Confidence Intervals Topic 33 (40) Estimating A Normal Population Mean μ (σ Known) A random sample of size 10 from a population of heights that has a normal distribution

### AP Statistics 2005 Scoring Guidelines

AP Statistics 2005 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check

### AP Statistics 2006 Scoring Guidelines

AP Statistics 2006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### High School Functions Building Functions Build a function that models a relationship between two quantities.

Performance Assessment Task Coffee Grade 10 This task challenges a student to represent a context by constructing two equations from a table. A student must be able to solve two equations with two unknowns

### AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Hypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7

Chapter 7 Hypothesis Testing with One Sample 71 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population

### Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?

Chapter 14: 1-6, 9, 1; Chapter 15: 8 Solutions 14-1 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np

### Constructing and Interpreting Confidence Intervals

Constructing and Interpreting Confidence Intervals Confidence Intervals In this power point, you will learn: Why confidence intervals are important in evaluation research How to interpret a confidence

### AP Statistics 2001 Solutions and Scoring Guidelines

AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use

### Section I: Multiple Choice Select the best answer for each question.

Chapter 15 (Regression Inference) AP Statistics Practice Test (TPS- 4 p796) Section I: Multiple Choice Select the best answer for each question. 1. Which of the following is not one of the conditions that

### Multiple Choice Questions

Multiple Choice Questions 1. A fast- food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant

### 4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Assume that the change in daily closing prices for stocks on the New York Stock Exchange is a random

### Confidence Intervals. Chapter 11

. Chapter 11 Confidence Intervals Topic 22 covers confidence intervals for a proportion and includes a simulation. Topic 23 addresses confidence intervals for a mean. Differences between two proportions

### Suppose we want to compare the average effectiveness of two treatments in a completely randomized experiment. In this case, the parameters µ 1

AP Statistics: 10.2: Comparing Two Means Name: Suppose we want to compare the average effectiveness of two treatments in a completely randomized experiment. In this case, the parameters µ 1 and µ 2 are

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### Statistics Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Statistics Final Exam Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Assume that X has a normal distribution, and find the indicated

### Hypothesis Testing. Bluman Chapter 8

CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

### Let m denote the margin of error. Then

S:105 Statistical Methods and Computing Sample size for confidence intervals with σ known t Intervals Lecture 13 Mar. 6, 009 Kate Cowles 374 SH, 335-077 kcowles@stat.uiowa.edu 1 The margin of error The

### Versions 1a Page 1 of 17

Note to Students: This practice exam is intended to give you an idea of the type of questions the instructor asks and the approximate length of the exam. It does NOT indicate the exact questions or the

### AP Statistics 2008 Scoring Guidelines Form B

AP Statistics 2008 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test.

Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations.

### CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN

CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN A. Introduction 1. this situation, where we do not know anything about the population but the sample characteristics, is far and away the most common circumstance

### Chapter 8: Hypothesis Testing of a Single Population Parameter

Chapter 8: Hypothesis Testing of a Single Population Parameter THE LANGUAGE OF STATISTICAL DECISION MAKING DEFINITIONS: The population is the entire group of objects or individuals under study, about which

### Confidence Intervals in Excel

Confidence Intervals in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### 12.1 Inference for Linear Regression

12.1 Inference for Linear Regression Least Squares Regression Line y = a + bx You might want to refresh your memory of LSR lines by reviewing Chapter 3! 1 Sample Distribution of b p740 Shape Center Spread

### ACTM Regional Statistics Multiple Choice Questions

ACTM Regional Statistics Multiple Choice Questions This exam includes 2 multiple- choice items and three constructed- response items that may be used as tie- breakers. Record your answer to each of the

### Instructor: Doug Ensley Course: MAT Applied Statistics - Ensley

Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics - Ensley Assignment: Online 15 - Sections 8.2-8.3 1. A survey asked the question "What do you think is the ideal number of children

### 6.1 The Elements of a Test of Hypothesis

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 22, 2012 Date of latest update: August 20 Lecture 6: Tests of Hypothesis Suppose you wanted to determine

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### 09 Confidence Intervals

09 Confidence Intervals 9.1 Inference and Point Estimates Whenever we use a single statistic to estimate a parameter we refer to the estimate as a point estimate for that parameter. When we use a statistic

### 1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700

Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,

### Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)

Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated

### Chapter (a) The sampling distribution of x is approximately normal with mean µ. standard deviation σ

3 Chapter. (a) The sampling distribution of x is approximately normal with mean µ x = 8 and σ 6 standard deviation σ x = =.7. (b) The mean is 8. One standard deviation from n 84 the mean: 77.9 and 8.;

### CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS

LESSON SEVEN CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS An interval estimate for μ of the form a margin of error would provide the user with a measure of the uncertainty associated with the point estimate.

### Why do we measure central tendency? Basic Concepts in Statistical Analysis

Why do we measure central tendency? Basic Concepts in Statistical Analysis Chapter 4 Too many numbers Simplification of data Descriptive purposes What is central tendency? Measure of central tendency A

### The calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n

EXAMPLE 1: Paired t-test and t-interval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel

### 1 Confidence intervals

Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.

### 9.2 Examples. Example 1

9.2 Examples Example 1 A simple random sample of size n is drawn. The sample mean,, is found to be 19.2, and the sample standard deviation, s, is found to be 4.7. a) Construct a 95% confidence interval

### Practice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.

Practice problems for Homework 1 - confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the

### CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Chi-Square. The goodness-of-fit test involves a single (1) independent variable. The test for independence involves 2 or more independent variables.

Chi-Square Parametric statistics, such as r and t, rest on estimates of population parameters (x for μ and s for σ ) and require assumptions about population distributions (in most cases normality) for

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 8.1 Homework Answers

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 8.1 Homework Answers 8.1 In each of the following circumstances state whether you would use the large sample confidence interval,

### Total Student Count: 3170. Grade 8 2005 pg. 2

Grade 8 2005 pg. 1 Total Student Count: 3170 Grade 8 2005 pg. 2 8 th grade Task 1 Pen Pal Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Convert cake baking temperatures

### ACTM State Exam-Statistics

ACTM State Exam-Statistics For the 25 multiple-choice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tie-breaker

### Problem Solving and Data Analysis

Chapter 20 Problem Solving and Data Analysis The Problem Solving and Data Analysis section of the SAT Math Test assesses your ability to use your math understanding and skills to solve problems set in

### 2002 AP Statistics Exam Multiple Choice Solutions

2002 AP Statistics Exam Multiple Choice Solutions 1. Answer: (D) (A) is FALSE There are not necessarily more subjects available for either experiments or observational studies. There are probably more

### STA 2023H Solutions for Practice Test 4

1. Which statement is not true about confidence intervals? A. A confidence interval is an interval of values computed from sample data that is likely to include the true population value. B. An approximate

### HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

### The Sampling Distribution of the Mean Confidence Intervals for a Proportion

Math 130 Jeff Stratton Name The Sampling Distribution of the Mean Confidence Intervals for a Proportion Goal: To gain experience with the sampling distribution of the mean, and with confidence intervals

### Statistics 2015 Scoring Guidelines

AP Statistics 2015 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### A Quick Guide to Confidence Intervals and Hypotheses Tests Using the TI-Calc AP Statistics

Example: Confidence Intervals for One Proportion In January 2007, Consumer Reports conducted a study of bacteria in frozen chicken sold in the US. They purchased a random selection of 525 packages of frozen

### MAT 118 DEPARTMENTAL FINAL EXAMINATION (written part) REVIEW. Ch 1-3. One problem similar to the problems below will be included in the final

MAT 118 DEPARTMENTAL FINAL EXAMINATION (written part) REVIEW Ch 1-3 One problem similar to the problems below will be included in the final 1.This table presents the price distribution of shoe styles offered

### A REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Performance Assessment Task Hexagons Grade 7 task aligns in part to CCSSM HS Algebra The task challenges a student to demonstrate understanding of the concepts of relations and functions. A student must

### Question 1 Question 2 Question 3 Question 4 Question 5 Question 6. Math 144 tutorial April, 2010

Math 144 tutorial 7 12 April, 2010 1. Let us define S 2 = 1 n n i=1 (X i X) 2. Show that E(S 2 ) = n 1 n σ2. 1. Let us define S 2 = 1 n n i=1 (X i X) 2. Show that E(S 2 ) = n 1 n σ2. 1. Let us define S

### Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

### Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

### AP Statistics 2007 Scoring Guidelines

AP Statistics 2007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### The Basics of a Hypothesis Test

Overview The Basics of a Test Dr Tom Ilvento Department of Food and Resource Economics Alternative way to make inferences from a sample to the Population is via a Test A hypothesis test is based upon A

### Math 201: Statistics November 30, 2006

Math 201: Statistics November 30, 2006 Fall 2006 MidTerm #2 Closed book & notes; only an A4-size formula sheet and a calculator allowed; 90 mins. No questions accepted! Instructions: There are eleven pages

### AP * Statistics Review. Probability

AP * Statistics Review Probability Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production of,

### Solutions 7. Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions.

Solutions 7 Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions. (1) Here we debunk a popular misconception about confidence intervals

### Chapter 6 Random Variables

Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

### STAT 3090 Test 3 - Version A Fall 2015

Multiple Choice: (Questions 1-16) Answer the following questions on the scantron provided using a #2 pencil. Bubble the response that best answers the question. Each multiple choice correct response is

### Chapter 23 Inferences About Means

Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

### Hints for Success on the AP Statistics Exam. (Compiled by Zack Bigner)

Hints for Success on the AP Statistics Exam. (Compiled by Zack Bigner) The Exam The AP Stat exam has 2 sections that take 90 minutes each. The first section is 40 multiple choice questions, and the second

### Inferences Based on a Single Sample Tests of Hypothesis

Inferences Based on a Single Sample Tests of Hypothesis 11 Inferences Based on a Single Sample Tests of Hypothesis Chapter 8 8. used to decide whether or not to reject the null hypothesis in favor of the