SOLUTIONSMAT 167: StatisticsTest I: Chapters 1-4Instructor: Anthony TanbakuchiFall 2008Name:Computer / Seat Number:No books, notes, or friends. Show your work. You may use the attachedequation sheet, R, and a calculator. No other materials. Write your work in theprovided space for each problem (including any R work if appropriate). You maynot use personal computers, only use the classroom computer at your desk. Usingany other program or having any other documents open on the computer willconstitute cheating.You have until the end of class to finish the exam, manage your time wisely.If something is unclear quietly come up and ask me.If the question is legitimate I will inform the whole class.Express all final answers to 3 significant digits. Probabilities should be given as adecimal number unless a percent is requested. Circle final answers, ambiguous ormultiple answers will not be accepted. Show steps where appropriate.The exam consists of 8 questions for a total of 40 points on 9 pages.This Exam is being given under the guidelines of our institution’sCode of Academic Ethics. You are expected to respect those guidelines.Points Earned: out of 40 total pointsExam Score:MAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 1 of 9Solution:Exam Results. Below are the summary statistics.> summary ( s c o r e )Min . 1 s t Qu. Median Mean 3 rd Qu. Max.30 . 00 67 . 50 81 . 25 75 . 11 85 . 62 92 . 50> par ( mfrow = c ( 1 , 2 ) )> box plo t ( sc o re , main = ”Boxplot o f exam s c o r e s ”)> h i s t ( s co re , main = ”Histogram o f exam s c o r e s ”)●30 50 70 90Boxplot of exam scoresHistogram of exam scoresscoreFrequency30 50 70 900 2 4 6 8 101. Provide short succinct written answers to the following conceptual questions.(a) (1 point) Would ethnicity be classified as a nominal, ordinal, interval, or ratio level ofmeasurement?Solution: Ethnicity would be categorical, with no meaningful order, therefore wewould classify it as nominal.(b) (1 point) Which of the following measures of center is least susceptible to outliers:median, mean, midrange, modeSolution: The mode is the least susceptible.(c) (1 point) What percent of data is greater than Q1?Solution: 75%(d) (1 point) If the mean, median, and mode for a data set are all the same, what can youconclude about the data’s distribution?Instructor: Anthony Tanbakuchi Points earned: / 4 pointsMAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 2 of 9Solution: If all three measures of center are the same, the distribution is symmetrical.(Not necessarily a normal distribution, all we know is that it is symmetrical.)(e) (1 point) If the mean is less than the mode for a data set, what can you conclude aboutthe data’s distribution?Solution: The data set is negatively skewed.(f) (1 point) What does the standard deviation represent conceptually in words? (Be concisebut don’t simply state the equation in words verbatim.)Solution: The standard deviation represents the average variation of the data fromthe mean.(g) (1 point) To determine the proportion of support in Tucson for Vice Presidential candidatePalin, a researcher randomly samples 10 people (using an ideal simple random sample).From the sample data, 70% expressed their support. The researcher was suspicious of theresult so she repeated the study randomly sampling 10 people 3 more times, the resultantstatistics were 30% 70% and 0% respectively. The researcher thinks some sort of mistakehas occurred causing the numbers to change each time the study is repeated. However,after carefully checking the procedures used, no mistake occurred. What type of errorcould we attribute to the variation between the sample results?Solution: This result should not be surprising, sampling error can cause fluctuationsin the statistic from sample to sample.(h) (2 points) A histogram is a useful tool that can quickly communicate many traits abouta set of data. List 4 useful pieces of information that an observer can easily assess usinga histogram.Solution: A histogram can be used to get an approximation of:1. central tendency2. variation in the data3. shape of the data4. assess if outliers exist5. min6. max(i) (1 point) Your SAT results state that you scored in the 95thpercentile. What does thismean?Solution: This is a good score, it means that 95% of the the people taking the testhad a lower score than you.(j) (1 point) Why would a SAT percentile be preferred over a raw SAT score for collegeadmissions committees?Instructor: Anthony Tanbakuchi Points earned: / 7 pointsMAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 3 of 9Solution: The percentile compares how the applicant did to their peers who took thetest (a measure of relative standing). A raw score doesn’t give information as to howthis score compared to others taking the test, making it hard to determine if a 1100is easy or hard to get.2. A survey conducted in our class asked 24 students how many credits they were enrolled in thissemester. Use the R output below to answer the following questions.There are 24 data points stored in the variable x, below is the sorted data:> s o r t ( x )[ 1 ] 6 10 10 10 11 11 11 12 12 12 12 12 12 13 13 13 14 15 15 16 16 16 17 19The basic descriptive statistical analysis is as follows:> summary ( x )Min . 1 s t Qu . Median Mean 3 rd Qu. Max .6 . 0 0 11 . 00 12 . 00 12 . 83 15 . 00 19 . 00> var ( x )[ 1 ] 7.8 84 058> sd ( x )[ 1 ] 2.8 07 856> h i s t ( x , xlab = ” c r e d i t s ” , main = ”Student Enrollment Data ”)Student Enrollment DatacreditsFrequency6 8 10 12 14 16 18 200 2 4 6 8(a) (1 point) Use the range rule of thumb to estimate the standard deviation. Is it close tothe actual standard deviation?Solution:> s . e s t = (max( x ) − min ( x ) ) / 4> s i g n i f ( s . est , 3)[ 1 ] 3 . 2 5Instructor: Anthony Tanbakuchi Points earned: / 1 pointsMAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 4 of 9The range rule is a rough estimate of σ, it is relatively close to the actual standarddeviation shown in the output.(b) (1 point) What is P25equal to?Solution: You can find the value in the summary output for the 1st quarter, it isequal to 11.(c) (1 point) What is the IQR (inter quartile range) equal to?Solution: IQR = Q3− Q1, using the summary output, the IQR is 4.(d) (1 point) What percent of the data falls within the IQR?Solution: The IQR contains the data between Q1and Q3or equivalently P25andP75, thus 50% of the data falls inside the IQR.(e) (1 point) What is the percentile for the