# Math 58. Rumbos Fall Solutions to Review Problems for Exam 2

Save this PDF as:

Size: px
Start display at page:

Download "Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2"

## Transcription

1 Math 58. Rumbos Fall Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable X. Explain your answer. (a) A fair coin is flipped ten times. Let X denote the number of times the coin comes up tails. Answer: Yes, X has a binomial distribution with parameters n = 10 and p = 1/2 because each flip is a Bernoulli trial with probability p = 1/2 and the flips are independent. (b) A fair coin is flipped multiple times. Let X denote the number of times the coin needs to be flipped until we see ten tails. Answer: No, X does not have a binomial distribution. First of all, there is no limit to the number of trials needed to see 10 tails. Thus, there is not fixed parameter n. Consequently, the values of X can range from 0, 1, 3, all the way to infinity. (c) A roulette wheel with one ball in it is turned six times. Let X denote the number of times the ball lands on red. Answer: Yes, X has a binomial distribution with parameters n = 6, and p, the probability that the ball will land on a red spot. The outcome of each spin of the roulette is independent from the outcome of any other spin. (d) There are ten people in the room: five men and five women. Three people are to be selected at random to form a committee. Let X denote the number of men on the three-person committee. Answer: No, X does not have a binomial distribution. X has a hypergeometric distribution with parameters the number of men (5), the number of women (5), and the number of members in each committee. The possible values of X are 0, 1, 2 and 3 and their corresponding probabilities are: ) ( 5 3) P (X = 0) = ( 5 0( 10 3 ) = 1 12 P (X = 1) = ( 5 ) ( 1 5 ( 2) 10 ) =

2 Math 58. Rumbos Fall and P (X = 2) = P (X = 1) = ( 5 ) ( 2 5 ( 1) 10 ) = ( 5 ) ( 3 5 ( 0) 10 ) = Note that these are different from the ones predicted by the binomial model; namely, P (X = k) = ( 3 k ) ( 1 2 ) k ( ) 3 k 1 for k = 0, 1, 2, 3. 2 For instance, ( ) ( ) P (X = 0) = = For each of the following scenarios, determine the appropriate distribution for the random variable X. (a) A fair die is rolled seven times. Let X denote the number of times we see an even number. Answer: X is binomial with parameters n = 7 and p = 1/2, the probability of a number from 1 to 6 coming our even. (b) A card is selected at random from a standard deck of shuffled cards. The color of the card is determined and the card is returned to the deck. The cards are shuffled again. This selection procedure is repeated sixty times. Let X denote the proportion of times the selected card is red. Answer: Let Y denote the number of cards out of the sixty that come out red. Then, X = Y is the mean of a random sample of n size n = 60 drawn from a Bernoulli distribution with parameter p = 1/2. By the Central Limit Theorem, the distribution of X is approximately normal with mean µ = p = 1/2 and variance p(1 p)/n, or 1/240; that is, ( ) 1 X is approximately N 2,

3 Math 58. Rumbos Fall (c) The percentage of female students at a large university is known to be 46%. A simple random sample of 100 students is to be taken. Let X denote the number of male students in the sample. Answer: X is binomial with parameters n = 100 and p = Alternatively, by the Central Limit Theorem, X approximately normal with mean 54 and variance (d) On any given Saturday during college football season, there are roughly 70 games being played. At each game, a fair coin is flipped to determine which team gets to kick off first. Let X denote the proportion of these coins that land heads. Answer: By the Central Limit Theorem, X is approximately normal with mean 1/2 and variance 1/ A set of ten cards consists of five red cards and five black cards. The cards are shuffled thoroughly. (a) Six of these cards will be selected at random. Let X denote the number of red cards observed in the set of six selected cards. Describe the probability distribution which appropriately models the random variable X. Answer: X has a hypergeometric distribution with parameters 5, 5 and 6; that is, ( 5 ( P (X = k) = k) 5 ) 6 k ( 10 ) for k = 0, 1, 2, 3, 4, 5. 6 (b) One card is to be selected at random. The color will be observed and the card replaced in the set. The cards are then thoroughly reshuffled. This selection procedure is repeated four times. Let X denote the number of red cards observed in these four trials. What is the mean of X? Answer: In this case, X has a binomial distribution with parameters n = 4 and p = 1/2. It then follows that the expected value of X is Determine whether each of the following statements is true or false. (a) The margin of error for a 95% confidence interval for the mean µ increases as the sample size increases.

4 Math 58. Rumbos Fall Answer: FALSE The margin of error for the mean is z σ n where z depends on the confidence level, σ is the standard deviation, and n is the sample size. Hence, as the sample size increases, the margin of error decreases. (b) The margin of error for a confidence interval for the mean µ, based on a specified sample size n, increases as the confidence level decreases. Answer: FALSE The margin of error for the mean is z σ n where z depends on the confidence level, σ is the standard deviation, and n is the sample size. The value z increases as the confidence level increases. Hence, the margin of error decreases as the confidence level decreases. (c) The margin of error for a 95% confidence interval for the mean µ decreases as the population standard deviation decreases. Answer: TRUE The margin of error for the mean is z σ n where z depends on the confidence level, σ is the standard deviation, and n is the sample size. Hence, the margin of error decreases as the population standard deviation decreases. (d) The sample size required to obtain a confidence interval of specified margin of error µ increases as the confidence level increases. Answer: TRUE The margin of error for the mean is z σ n

5 Math 58. Rumbos Fall where z depends on the confidence level, σ is the standard deviation, and n is the sample size. If we call the margin of error ME, we obtain that ME = z σ n from which we get that ( n = z σ ) 2. ME As the confidence level increases, so does z, and consequently n also increases. 5. Certain Middle School has calculated a 95% confidence interval for the mean height µ of 11 year old boys at their school and found it to be 56 ± 2 inches. Determine whether each of the following statements is true or false. (a) There is a 95% probability that µ is between 54 and 58. Answer: False Confidence intervals for the population mean, µ, for the cases in which the population standard deviation, σ, is known, are computed from the sampling distribution of the sample mean, X n. For large sample sizes, n, we can use the Central Limit Theorem to approximate P (X n z n σ < µ < X n + z n σ ) P ( z < Z < z ), where Z is the standard normal random variable. Thus, the value P ( z < Z < z ) approximates the probability that if we collect a simple random sample of size n from the population, compute the sample mean, X n from that sample, and then form the interval (X n z σ n, X n + z σ n ), the interval contains the true population mean. Thus, a 95% confidence interval (54, 58) does not mean that there is a probability of 95% that µ will be in that interval. It means, according to Moore, McCabe and Craig (on page 367 of the sixth edition of Introduction to the Practice of Statistics), that the interval was calculated by a method that gives correct results in

6 Math 58. Rumbos Fall % of all possible samples. It does not say that the probability is 95% that the true mean lies between 54 and 58. No randomness remains after we draw a particular random sample and compute the interval. The true mean either is or is not between 54 and 58. The probability calculations of standard statistical inference describes how often the method, not the particular sample, gives the correct answer. (b) There is a 95% probability that the true mean is 56, and there is a 95% chance that the true margin of error is 2. Answer: False A 95% confidence interval means that the procedure used to generate the interval will capture the true mean approximately 95% of the time. (c) If we took many additional random samples of the same size and from each computed a 95% confidence interval for µ, approximately 95% of these intervals would contain µ. Answer: True This is the definition of a 95% confidence interval. (d) If we took many additional random samples of the same size and from each computed a 95% confidence interval for µ, approximately 95% of the time µ would fall between 54 and 58. Answer: False Each time we collect a sample of size n and compute the sample mean and corresponding interval, we will obtain in general values different from 54 and A nationally distributed college newspaper conducts a survey among students nationwide every year. This year, responses from a simple random sample of 204 college students to the question About how many CDs do you own? resulted in a sample mean X n = Based on data from previous years, the editors of the newspaper will assume that σ = 7.2. (a) Use the information given to obtain a 95% confidence interval for the mean number of CDs owned by all college students. Solution: Use the formula (X n z σ n, X n + z σ n ),

7 Math 58. Rumbos Fall where z = 1.96 and n = 204 to obtain (71.8, 73.8) (b) Answer each of the following questions with yes, no, or can t tell. i. Does the sample mean lie in the 95% confidence interval? Answer: Yes. ii. Does the population mean lie in the 95% confidence interval? Answer: Can t tell. iii. If we were to use a 92% confidence level, would the confidence interval from the same data produce an interval wider than the 95% confidence interval? Answer: No. In this case, z is about So, the interval is shorter. iv. With a smaller sample size, all other things being the same, would the 95% confidence interval be wider? Answer: Yes. As n decreases, the margin of error, z interval will be wider. σ n, increases; so, the 7. A review of voter registration records in a small town yielded the following table of the number of males and females registered as Democrat, Republican, or some other affiliation: Affiliation \ Gender Male Female Democrat Republican Other Table 1: Data for Problem 7 Suppose we wish to test the null hypothesis that there is no association between party affiliation and gender. Under the null hypothesis, what is the expected number of male Democrats? Solution: Compute the row totals, column totals and grand total as shown in Table 2.

8 Math 58. Rumbos Fall Affiliation \ Gender Male Female Row Totals Democrat Republican Other Column Totals Table 2: Two way table for Problem 7 The expected values under the assumption of no association between the variables are then computing my multiplying row and column totals and dividing by the grand total. The results are shown in Table 3. Affiliation \ Gender Male Female Row Totals Democrat Republican Other Column Totals Table 3: Expected values for Problem 7 Thus, the expected number of male democrats would be A specific type of electronic sensor has been experiencing failures. The manufacturer has studied 200 of these devices and has determined the type of failure (A, B, C) that occurred and the location on the sensor where the failure happened (External, Internal). The following table summarizes the findings: Location \ Type A B C Total Internal External Total Table 4: Data for Problem 8 Test the null hypothesis is that there is no association between Type of Failure and Location of Failure. Solution: We present the solution to this problem using a permutation procedure in R.

9 Math 58. Rumbos Fall First we enter the counts in the two way table as a 2 3 matrix which we call Obs. Type Obs <- matrix(c(50,16,31,61,26,16), 2,3,byrow=T) to get the output > Obs [,1] [,2] [,3] [1,] [2,] The expected values, assuming that there is no association between the type of failure and location of failure, are obtained by applying the following function which we call expected() defined by the following code: # Computing expected values for 2-way tables # Enter M: the matrix of counts in 2-way table expected <- function(m) { R <- matrix(rowsums(m)) C <- matrix(colsums(m),1,dim(m)[2]) E <- (R %*% C)/sum(M) return(e) } Typing Exp <- expected(obs), we obtain > Exp [,1] [,2] [,3] [1,] [2,] The Chi Squared distance is then obtained from Xsqr <- sum((obs-exp)^2/exp)}. This yields a value of about

10 Math 58. Rumbos Fall To estimate the p value for the test, we run the IndepTest() function defined by the following code: # Permutation Test of Independence for 2-way tables # # Enter a matrix, M, of observed counts in 2-way table # and the number of repetitions, N # This code assumes that the expected() function has # been defined. # IndepTest <- function(m,n) { Exp <- expected(m) # computes expected values CS <- array(dim=n) # sets up array dimension N RowGroups <- rep(1:dim(m)[1],rowsums(m)) # sets up row groups L <- length(rowgroups) ColGroups <- rep(1:dim(m)[2],colsums(m)) # sets up column groups for (k in 1:N) { perms <- sample(rowgroups,l) Sim <- table(perms,colgroups) CS[k] <- sum((sim-exp)^2/exp) } return(cs) } Typing ChiSqr <- IndepTest(Obs,10000) yields the array ChiSqr of 10, 00 simulated Chi Squared distances. The p value can then be estimated by p value phat(chisqr,xsqr) Hence, the p value is about 1.85%. Thus, we can reject the null hypothesis at the 5% significance level. We can then say that there is strong evidence to suggest that there is an association between the type of failure and the location.

11 Math 58. Rumbos Fall Alternate Solution: We can also estimate the p value in the previous solution by using the χ 2 df distribution with df = (r 1)(c 1) degrees of freedom. In this case r = 2 and c = 3, so that df = 2. The estimate for the p value is then p value P (χ 2 2 Xsqr). In R, we may estimate this further as p value 1-pchisq(Xsqr,2) We deduce the same conclusion as before. 9. A researcher is interested in determining if the model used for the distribution of main economic concerns in the year 2003 for residents in a certain county can still be used in the year A sample of 370 residents from that county was surveyed in The following table displays the model for the distribution of economic concerns for the year 2003 and the observed number of sampled respondents in the survey for the same economic concerns for the year 2004: Jobs Medical Care Higher Education Housing 2003 Model 5% 36% 27% 32% 2004 Counts Table 5: Data for Problem 9 Perform an appropriate significance test. Solution: We perform a goodness of fit test with the null hypothesis that the observations come from a population whose probability distribution is given by the proportions in the 2003 Model: H o : p 1 = 0.05, p 2 = 0.36, p 3 = 0.27, p 4 = We enter these proportions in an array called Prop in R: Prop <- c(0.05, 0.36, 0.27, 0.32) The observed counts are entered in a vector called Obs:

12 Math 58. Rumbos Fall Obs <- c(25, 138, 116, 91) The total number of observations, or the sample size, is then n <- sum(obs) which yields n = 370. The expected values are then Exp <- n*prop This yields > Exp [1] The Chi Squared distance is then given by Xsqr <- sum((obs-exp)^2/exp) which yields > Xsqr [1] At this point we may perform a randomization test to simulate taking samples of size n from a population with a distribution given by the null hypothesis. We present here a code that can be used for all goodness of fit tests like this one. It takes the vector of observed counts, the vector of postulated proportions, and the desired number of repetitions. # Randomization Test for Goodness of Fit # Author: Adolfo J. Rumbos # Date: December 3, 2008 # # Enter an array of observed counts, Obs, and array of postulated # proportions, P, and the number of repetitions, N # GoodFitTest <- function(obs,p,n)

13 Math 58. Rumbos Fall { k <- length(obs) # number of categories n <- sum(obs) # sample size Exp <- n*p # computes expected values CS <- array(dim=n) # sets up array dimension N CatCounts <- round(100*prop) # Expected counts per categories in 100 Categories <- rep(1:k,catcounts) for (i in 1:N) { s <- sample(categories,n,replace=t) # simulated samples Sim <- table(s) CS[i] <- sum((sim-exp)^2/exp) } return(cs) } Typing ChiSqr <- GoodFitTest(Obs,Prop,10000) returns an array of 10, 000 simulated values for the Chi Squared distance. The histogram of this particular run is shown in Figure 1. Histogram of ChiSqr Frequency ChiSqr Figure 1: Histogram of ChiSqr We now estmate the p value as usual, applying the phat() function with the observed Chi Squared distance, Xsqr, as input to obtain a

14 Math 58. Rumbos Fall p value of about Thus, at the 5% level of significance, we can reject the null hypothesis. Alternate Solution: As was done in the previous problem, we can also estimate the p value by using the χ 2 df distribution with df = k 1 = 3 degrees of freedom. The estimate for the p value is then p value P (χ 2 3 Xsqr). In R, we may estimate this further as p value 1-pchisq(Xsqr,3) We deduce the same conclusion as before. Perhaps, we could claim a 1% significance level; but then again, this is another approximation to the p value. 10. There have been many studies that looked at the incidence of heart attack on the different days of the week. Studies in Japan and Scotland seemed to find that there was a substantial spike in heart attack on Mondays, perhaps as many as 20% more. A researcher studied a random selection of 200 heart attack victims and recorded the day of the week that their attack occurred. The following table summarizes the results: Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday Counts Table 6: Data for Problem 10 The researcher was interested in whether the distribution of heart attacks was the same or different across the days of the week. Perform an appropriate significance test to answer this question. Solution: We perform a goodness of fit test. In this case, Obs <- c(24, 36, 27, 26, 32, 26, 29) n <- sum(obs)

15 Math 58. Rumbos Fall k <- length(obs) Prop <- rep(1/k,k) Exp <- n*prop Xsqr <- sum((obs - Exp)^2/Exp) The last line yields a value for Xsqr of We estimate the p value by performing a randomization test using the GoodFitTest() function defined in the previous problem: ChiSqr <- GoodFitTest(Obs,Prop,10000) The density histogram of ChiSqr is shown in Figure 2. In this case the Desnsity Histogram of ChiSqr Density ChiSqr Figure 2: Density Histogram of ChiSqr p value is about , and so we cannot reject the null hypothesis. Thus, the data do not support the assertion that distribution of heart attacks was different across the days of the week. Thus, the day of

16 Math 58. Rumbos Fall the week does not have any bearing of the number of heart attacks.

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of

### The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

### Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

### Elementary Statistics

lementary Statistics Chap10 Dr. Ghamsary Page 1 lementary Statistics M. Ghamsary, Ph.D. Chapter 10 Chi-square Test for Goodness of fit and Contingency tables lementary Statistics Chap10 Dr. Ghamsary Page

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Chi Square for Contingency Tables

2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure

### Mind on Statistics. Chapter 15

Mind on Statistics Chapter 15 Section 15.1 1. A student survey was done to study the relationship between class standing (freshman, sophomore, junior, or senior) and major subject (English, Biology, French,

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### 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

### X: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001

Tuesday, January 17: 6.1 Discrete Random Variables Read 341 344 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

### Statistics 151 Practice Midterm 1 Mike Kowalski

Statistics 151 Practice Midterm 1 Mike Kowalski Statistics 151 Practice Midterm 1 Multiple Choice (50 minutes) Instructions: 1. This is a closed book exam. 2. You may use the STAT 151 formula sheets and

### STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS

STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### AP Statistics for Friday 4/15/16. 2) turn in work; hand back work. 3) lesson on Chi Square tests for Association with handouts

AP Statistics for Friday 4/15/16 1) go over schedule 2) turn in work; hand back work 3) lesson on Chi Square tests for Association with handouts 4) Assign #105 p 631 / 19, 23, 31 schedule... Normal Distribution

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### Math 58. Rumbos Fall Solutions to Assignment #5

Math 58. Rumbos Fall 2008 1 Solutions to Assignment #5 1. (Typographical Errors 1 ) Typographical and spelling errors can be either nonword errors or word errors. A nonword error is not a real word, as

### 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

### Thursday, November 13: 6.1 Discrete Random Variables

Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

### Chi Square Distribution

17. Chi Square A. Chi Square Distribution B. One-Way Tables C. Contingency Tables D. Exercises Chi Square is a distribution that has proven to be particularly useful in statistics. The first section describes

### Name: Date: Use the following to answer questions 2-3:

Name: Date: 1. A study is conducted on students taking a statistics class. Several variables are recorded in the survey. Identify each variable as categorical or quantitative. A) Type of car the student

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Review #2. Statistics

Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of

### 6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

### Math 58. Rumbos Fall 2008 1. Solutions to Assignment #3

Math 58. Rumbos Fall 2008 1 Solutions to Assignment #3 1. Use randomization to test the null hypothesis that there is no difference between calcium supplementation and a placebo for the experimental data

### 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.

### CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### 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

### Is it statistically significant? The chi-square test

UAS Conference Series 2013/14 Is it statistically significant? The chi-square test Dr Gosia Turner Student Data Management and Analysis 14 September 2010 Page 1 Why chi-square? Tests whether two categorical

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### CHAPTER IV FINDINGS AND CONCURRENT DISCUSSIONS

CHAPTER IV FINDINGS AND CONCURRENT DISCUSSIONS Hypothesis 1: People are resistant to the technological change in the security system of the organization. Hypothesis 2: information hacked and misused. Lack

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Chi Square Test. Paul E. Johnson Department of Political Science. 2 Center for Research Methods and Data Analysis, University of Kansas

Chi Square Test Paul E. Johnson 1 2 1 Department of Political Science 2 Center for Research Methods and Data Analysis, University of Kansas 2011 What is this Presentation? Chi Square Distribution Find

### Test of proportion = 0.5 N Sample prop 95% CI z- value p- value (0.400, 0.466)

STATISTICS FOR THE SOCIAL AND BEHAVIORAL SCIENCES Recitation #10 Answer Key PROBABILITY, HYPOTHESIS TESTING, CONFIDENCE INTERVALS Hypothesis tests 2 When a recent GSS asked, would you be willing to pay

### Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 7 Estimates and Sample Sizes 7 1 Review and Preview 7 2 Estimating a Population Proportion 7 3 Estimating a Population

### Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

### 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:

### Problem sets for BUEC 333 Part 1: Probability and Statistics

Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are back-of-chapter exercises from

### 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

### E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

### 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)

### statistics Chi-square tests and nonparametric Summary sheet from last time: Hypothesis testing Summary sheet from last time: Confidence intervals

Summary sheet from last time: Confidence intervals Confidence intervals take on the usual form: parameter = statistic ± t crit SE(statistic) parameter SE a s e sqrt(1/n + m x 2 /ss xx ) b s e /sqrt(ss

### Chapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics

Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3

### Least Squares Estimation

Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science

STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto

### MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### N Mean Std. Deviation Std. Error of Mean

DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 815 SAMPLE QUESTIONS FOR THE FINAL EXAMINATION I. READING: A. Read Agresti and Finalay, Chapters 6, 7, and 8 carefully. 1. Ignore the

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### In the past, the increase in the price of gasoline could be attributed to major national or global

Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

### Chi-square test Testing for independeny The r x c contingency tables square test

Chi-square test Testing for independeny The r x c contingency tables square test 1 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided

### Unit 12 Logistic Regression Supplementary Chapter 14 in IPS On CD (Chap 16, 5th ed.)

Unit 12 Logistic Regression Supplementary Chapter 14 in IPS On CD (Chap 16, 5th ed.) Logistic regression generalizes methods for 2-way tables Adds capability studying several predictors, but Limited to

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.

Probability and the Chi-Square Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system

### Odds ratio, Odds ratio test for independence, chi-squared statistic.

Odds ratio, Odds ratio test for independence, chi-squared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review

### fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson

fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson Contents What Are These Demos About? How to Use These Demos If This Is Your First Time Using Fathom Tutorial: An Extended Example

### Stats Review Chapters 9-10

Stats Review Chapters 9-10 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

### STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

### Testing Hypotheses using SPSS

Is the mean hourly rate of male workers \$2.00? T-Test One-Sample Statistics Std. Error N Mean Std. Deviation Mean 2997 2.0522 6.6282.2 One-Sample Test Test Value = 2 95% Confidence Interval Mean of the

### PASS Sample Size Software

Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### 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

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540

Probability, Binomial Distributions and Hypothesis Testing Vartanian, SW 540 1. Assume you are tossing a coin 11 times. The following distribution gives the likelihoods of getting a particular number of

### 12: Analysis of Variance. Introduction

1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider

### Statistical Functions in Excel

Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

### Simulating Chi-Square Test Using Excel

Simulating Chi-Square Test Using Excel Leslie Chandrakantha John Jay College of Criminal Justice of CUNY Mathematics and Computer Science Department 524 West 59 th Street, New York, NY 10019 lchandra@jjay.cuny.edu

### 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

### 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.

### Measuring the Power of a Test

Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection

### \$2 4 40 + ( \$1) = 40

THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups

### Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

### 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

### LOGNORMAL MODEL FOR STOCK PRICES

LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

### Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams

Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set

### Homework 5 Solutions

Math 130 Assignment Chapter 18: 6, 10, 38 Chapter 19: 4, 6, 8, 10, 14, 16, 40 Chapter 20: 2, 4, 9 Chapter 18 Homework 5 Solutions 18.6] M&M s. The candy company claims that 10% of the M&M s it produces

### Having a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.

Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

### Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Open book and note Calculator OK Multiple Choice 1 point each MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data.

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### CHI SQUARE DISTRIBUTION

CI SQUARE DISTRIBUTION 1 Introduction to the Chi Square Test of Independence This test is used to analyse the relationship between two sets of discrete data. Contingency tables are used to examine the

Summer 2013 Term June 3, 2013 (Monday) June 3-4, 2013 (Monday Tuesday) June 5, 2013 (Wednesday) June 5-6, 2013 (Wednesday Thursday) June 6, 2013 (Thursday) July 3, 2013 (Wednesday) July 4, 2013 (Thursday)

### Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

### AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men