SOLUTIONSMAT 167: StatisticsTest I: Chapters 1-4Instructor: Anthony TanbakuchiSpring 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 10 questions for a total of 46 points on 10 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 46 total pointsExam Score:MAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 1 of 10Solution:Exam Results. Below are the summary statistics.> summary ( s c o r e )Min . 1 s t Qu. Median Mean 3 rd Qu. Max.20 . 00 60 . 00 75 . 56 71 . 53 87 . 78 10 0. 00> 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 ”)20 40 60 80 100Boxplot of exam scoresHistogram of exam scoresscoreFrequency20 40 60 80 1000 1 2 3 4 51. Provide short succinct written answers to the following conceptual questions.(a) (1 point) Would mass in grams be classified as a nominal, ordinal, interval, or ratio levelof measurement?Solution: Mass is a ratio level of measurement since 0 mass means there is no massat all.(b) (1 point) Which of the following measures of variation is most effected by outliers:IQR, standard deviation, rangeSolution: The range is most effected by outliers(c) (1 point) What percent of data is greater than Q3?Solution: 25%(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 10Solution: 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 median is greater than the mode for a data set, what can you concludeabout the data’s distribution?Solution: The data set is positively 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) Give an example of sampling error?Solution: Use a simple random sample of 5 students from a class to estimate themean weight of students in the class. Each time you repeat the process, you don’tnecessarily get the same sample mean because you don’t always have the same peoplein the sample.(h) (2 points) A box plot is a useful tool that can quickly communicate many traits about aset of data. List 4 useful pieces of information that an observer can easily assess using abox plot.Solution: A box plot can be used to get an approximation of:1. central tendency2. variation in the data3. IQR, Q1, Q2, median4. shape of the data5. assess if outliers exist6. min7. max(i) (1 point) You scored in the 78thpercentile on the GRE. If 8,000 people took the GRE,how many people had a score at least as high as your score?Solution: If you were in the 78th, then 78% of the people scored lower than you and22% did at least as well as you. Therefore, 8000 · 0.22 = 1760 people who had a scoreat least as high as yours.2. A survey conducted in our class asked 18 students how many hours they work per week. Usethe R output below to answer the following questions.There are 18 data points stored in the variable x, below is the sorted data:Instructor: Anthony Tanbakuchi Points earned: / 6 pointsMAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 3 of 10> s o r t ( x )[ 1 ] 0 0 0 0 8 8 10 12 13 20 21 30 30 30 30 30 35 50The basic descriptive statistical analysis is as follows:> summary ( x )Min . 1 s t Qu. Median Mean 3 rd Qu. Max.0 . 0 0 8 . 0 0 16 . 50 18 . 17 30 . 00 50 . 00> var ( x )[ 1 ] 215 .6 765> sd ( x )[ 1 ] 14. 68 593> h i s t ( x , xlab = ”hours ” , main = ”Student Enrollment Data ”)Student Enrollment DatahoursFrequency0 10 20 30 40 500 1 2 3 4 5 6 7(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 ] 1 2 . 5The 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 P50equal to?Solution: You can find the value in the summary output for the 1st quarter, it isequal to the median 16.5.(c) (1 point) What is the IQR (inter quartile range) equal to?Instructor: Anthony Tanbakuchi Points earned: / 3 pointsMAT 167: Statistics, Test I: Chapters 1-4 SOLUTIONS p. 4 of 10Solution: IQR = Q3− Q1, using the summary output, the IQR is 22.(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 mode for the data?Solution: The mode is 30 hours per week (it occurs 5 times).(f) (1 point) What is the percentile for a student who works 8 hours per week?Solution: The percentile is the percent of data points less than the given data point.A student working 8 hours per week could be position 5 or 6, use i = 5.5, thus:Pk= xi, (sorted x)k =i − 0.5n· 100%=5.5 − 0.518· 100%= 27.8%(g) (1 point) What is the z-score for the student who is working 50 hours per week?Solution:z =x − ¯xs> x . bar[ 1 ] 18. 16 667> s[ 1 ] 14. 68 593> z = (50 − x . bar )/ s> s i g n i f ( z , 3)[ 1 ] 2 . 1 7(h) (1 point) Is 35 hours an unusual (outlier) value based on its z score? (Why)Solution:> x . bar[ 1 ] 18. 16 667> sInstructor: Anthony …