Stat 110 رياضيات واحصاء السنة التحضيرية Ch 3.

Transcription

1 Stat 110 Ch 3 محمد عمران السنة التحضيرية رياضيات واحصاء

2 Data description Measures of central tendency Or measures of average هقا س ال ضعة الوشكض ة A statistic A parameter اإلحصائ المعلمة هو مق اس صف خاص ة من خصائص الع نة )sample( هو مق اس صف خاص ة من خصائص المجتمع )population( Type 1-mean (arithmetic average) of measure الوسط الحسابى In most case is not an actual ل س بالضرورة ان كون الوسط الحساب أحد ق م الب انات المعطاة Properties of mean 1) The mean computed by using all the values of a data set 2) The mean for data set is unique and not necessarily of the data value 3) The mean cannot be computed for open- ended frequency distribution 4) The mean is affected by extremely high or low values and may not be the appropriate average الوسط الحساب تأثر بالق م الكب رة جدا وبالق م الصغ رة جدا )الق م الشاذة( For sample x x n For population x n 1

3 Example Find mean for the data 1)

4 ال س ط ) (MD Median Median is midpoint of the data array Properties of median 1) The median is used when find the center or middle value of a date must set 2) The median is used when one must determine whether the data values into the upper half or the lower half of the distribution ستخذم ال س ط لتحذ ذ ل الب ا ات تقع ف ال صف العل هي الت ص ع أ ال صف السفل هي الت ص ع 3) The median is used to find the average of an open ended distribution 4) The median is affected less than the mean by extremely high or extremely law value Note قط الو تصف لوجو ع الب ا ات When the distribution is extremely skewed the median rather than the mean is more a pproprian measure of central tendency الوس ط اقل تأثرا بالق م الكب ر هاو الق م الصغ رة )الشاذة( من الوسط الحساب طش ق الحص ل عل ال س ط 1( شتب القشاءات تشت با تصاعذ ا ا ت اصل ا 2( ب ذاء بحزف األ ل هع األخ ش ثن التال هي ا هع التال هي اك إرا تبق عذد احذ ف ال س ط إرا تبق عذد ي ف جوع ن قسو ن عل 2 ال اتج ال س ط 3

5 Example Find median of the data 1)713,300,618,595,311,401,202 الترتيب 202,300,311,401,595,618,713 Medain =401 2)684,764,656,702,856,1132,1174,1199 التشت ب 656,684,702,764,856,1132,1174,1199 median

6 Mode Mode is the value that occurs most often in a dataset Properties 1) The mode is the only measure of central tendency that can be used when the data are nominal or categorical 2) The mode is not always unique 3) The mode is UN model ف حاله إن المنوال له ق مه واحده 4) The mode is bimodal 5) The mode is multimodal 6) The mode is no mode ف حاله إن المنوال له ق متان ف حاله إن المنوال له أكثر من ق متان ف المنوال حاله انه ال وجد منوال تسمى Example Find the mode of data 1)110,731,1031,84,20,118,1102,1977,103 Mode = no mode 2)8,9,914,8,8,10,7,6,9,7,8,10,11,8,14,11 Mode = 8 unmodel 3)104,104,104,104,104,107,109,109,109,109,109,109,111,111,112 Mode =109 unmodel 5

7 Distribution shapes إشكال التوز عات Symmetric distribution 1) Mean=median=mode التوز ع منتظم 2) Mode median mean ملتو الى ال م ن skewed) positively skewed (right Mean mode = + 3) Mean median mode ملتو الى ال سار skewed) negatively skewed (left Mean mode=- 6

8 Example 1) If the mean=median=mode=5, then the distribution is (symmetric) 2) If the mode= 5 and mean =6 then the distribution is (right or positive skewed ) 3) If the mode =5 and the mean = 4 then the distribution (left or negative skewed) The mid rang 1) The mid-range is easy to compute 2) The mid-range gives the mid-point 3) The mid-range is affected by extremely high or low values in data set Mid rang = Example 1) Find the mid rang for the data 2, 3,6,8,4 2) The values of the median for the values -3,1,-1,-5,-1,-7,1 is a)-1 b)-5 c)1 d)5 sol الترت ب -7,-5,-3,-1,-1,1,1 medain is -1 7

9 3) The raw data set for the stem and leaf plot shown is called a)uni model b)bi model c)multimodal 4) If the number of data set is 7 and it is mean is 8. Then the sum of data values is a) 40 b) 60 c) 80 d) 56 SOL 8

10 5) If the marks of three students are 6, X, 8 and the mean of these marks is 8 then X is a) 8 b) 10 c) 7 d) 9 sol 6) Half of observation are always greater than the a) Median b) mean c) mod d) mid rang 7) Which measure of central tendency are not affected by largely by outliers A) Mid-range and mod b) Median and mod c) Median and mean 8)..is not measure of central tendency a) Mean b) range c) median d) mode 9) A measure obtained from sample data is called a A) Statistic b) population c) parameter 10) What is the term for characteristic measure obtained by using all data values for a specific population? A) Variable b) mode c) statistic d) parametric 9

11 11) If the distribution is symmetric and it is mean equal 10, then the mode equals a) 10 b) 10 c) more than 10 12) If the mode is to the left of the median. And the mean is to the right of the median, then the distribution is a) Symmetric b) right skewed c) left skewed d) negatively skewed 13) The use all data to compute it a) Mean b) range c) mode d) median 14) If each value in a sample has the same frequency, then the data a) Are bimodal b) are multi model C) Have no mode d) are UN model 15) What is the measure of central tendency that appropriate to use when distribution is extremely skewed a) Mean b) median c) mid-range 16) When data are categorize, the most appropriate measure of central tendency? a) Mean b) mode c) median 17) What is the value of the mode, when all values in the data set are different? a) 0 b) 1 c) no mode 18) The symbol for the population mean is A) x b) c) M D d) 10

12 The weight mean w x w x... w nx w w... w 1 2 Where w 1, w 2, w 3,...w n are weight And x 1, x 2, x 3,... x n are values n n = wx w Example 1) A student received an A in English (3 credits).c in statistics (3 credits). a B in physics (4 credit) and a D in mathematics ( 2credits ) assuming A = 4 grade point B = 3grade points C =2 grade points D=1 grades points Find the students grade point average? W X Weight Grade English A 3 4 Stat C 3 2 Phy B 4 3 math D 2 1 ( ( ( ( 11

13 2)an instructor grades exams 20%,term paper 30%,final exam 50% a student has grade of 83,72 and 90 respectively, for exams,term paper and final exam.find the weight mean? W X ( ( ( 12

14 Measure of variation مقا س التشتت Or measure of dispersion (determine the spread of data value) 1)Rang ( R ) R = highest value lowest value 2) Variance and standard deviation For population Variance = = mean 2 X Standard deviation = For sample 2 Variance = s = = x 2 X n X = x X 2 n 1 X = x 2 n X x 2 x 2 n n 1 Standard deviation = S = x X 2 n = x 2 x 2 n n 1 13

15 3) The coefficient of variation For sample For population c var s = X.100% X = X.100% c var مالحظه هامه إذا كان معامل االختالف لمجتمع ما )أو عيىه( أكبر مه معامل االختالف لمجتمع اخر أو )عيىه أخرى( فان المجتمع األول )العينة األولى( أكثر تغيرا If c var 1 c var 2 Then the first is more variable than second Example If the mean of test A is 70 and it is standard deviation is 7. The mean of test B is 75 and it is standard deviation is 8 which is more variable test A or test B? 14

16 If the variance of the distribution is 16 then the standard deviation of the distribution is a) 16 b) 8 c) 4 d) 1 sol A measure used to compare the variation of two sets of data is called A) Coefficient of variation b) range c) Mean d) Mode Example The number of high way miles per gallon of the ten (10) worst vehicles shown 12,15,13,14,15,16,17,16,17,18 Find 1) Mean 2) median 3) mod 4) mid rage 5) range 6) variance 7) Standard deviation 8) coefficient of variation الترتيب 15

17 ( 16

18 Percentile percential للحصول على p 1 -نرتب االعداد ترت ب تصاعدي ثم نعوض بالقانون التالى (number of value below x total number of value Example A teacher gives a 20-point test to 10 students. The scores are shown here. 18, 15, 12,6,8,2,3,5,20,10 1)Find the percentile of a score 12 التشت ب 2,3,5,6,8,10,12,15,18,20 2) Find the percentile rank for a score 6 17

19 إذا اعطانا النسبة p وطلب منا الرتبة c نعوض في القانون c n p 100 وإذا كان الناتج عدد نسبى )به فواصل( تقريبه الى اقرب عدد صحيح العدد العدد التالى وإذا كان الناتج عدد صحيح فالرتبة = 2 Example Using the data to get the value that corresponds to the 1)25th percentile ( الىاتج عدد وسبى العذد الز سبت % 25 ستبت العذد الثالث= 5 25th 2)60th percentile عدد صحيح ( 2,3,5,6,8,10,12,15,18,

20 Z score or standard score Measures of position مقا س الموضع Z = For sample For population Z = x x s x X Z Score tell how many standard deviation of the data value is above below the mean Ex Z = Find Z score for each test A and test B which is higher Test A X= 38 x = 40 S =50 Test B X=94 x = 100 S= 10 X 19

21 The mean mark in test A is 80 and the standard deviation is 5. If a student get 70 in the test. His relative position (z- score) in the test is a) 6 b) 4 c) 3 sol d)-2 If the mean of a set of data is 24 and 18.4 has a (z-score) = -1.4 then the standard deviation must be a) 8 b) 2 c) 16 d) 4 sol Note 1-if the z score is positive x x 2-if the z- score is zero z=0 x x 3-if the z-score is negative x x 20

22 A box plot Exploratory data analysis استطالع تحل ل الب ا ات 1) The lowest value of the data 2) 3) median 4) 5) the heights value of the data تسو ز الق ن هجو ع الب ا ات Five number summary of the data set Is a median Q 2 Inter quartile rang I Q Q Q R = 3 1 min max 25% 25% 25% 25% Example Construct a box plot for the data 33,38,43,30,29,40,51,27,23,31, Q 1 Q 3 Min Q 2 Max

23 Find Q, Q, Q for the data set ,13,6,5,12,50,22,18 Q Find Q, Q, Q for the data set , 18, 22,19,3,21,17,20 Q

24 Outliers 1) Find Q 1 Q 2 Q 3 2) find I QR Q 3 Q 1 3) Find Q 1 (1 5 IQR Q 3 (1 5 IQR 4) The low value of Q 1 (1 5 I QR and height value of Q 3 (1 5 IQR called outliers Find outliers for data set 24,32,54,31,16,18,19,14,17,20 Example 1 Q ( ( ( ( ( ( ( 23

25 All value in a data set are between 6 and 15, expect for one value of 85 the value 85 is likely to be A) The rang b) the box plot c) the mean d) outliers Use the box plot to identify the maximum value, minimum, median, first quartile.third quartile, and interquartile range

26 Note box The distribution is approximately symmetric Symmetric 1( إرا كاى ال س ط قش با هي هشكض إل كاى ال س ط إل و ي هشكض إل box The distribution is negatively skewed (left skewed) 2( إرا Left skewed كاى ال س ط إل ساس هشكض إل box The distribution is positively skewed (right skewed) 3( إرا Right skewed 25

27 Use the box plot to indent,, MD, IQR, min value, max value, and show that if this distribution is QQ 1 3 positively or negatively SOL 26

28 Test bank 1- A characteristic or measure obtained by using the data values from a population is called a. A. Statistic B. Quartile C. Percent D. Parameter 2-.. is the symbols of mean in the sample A. B. σ² C. μ D. S² 3- Find the mean of following data 10,15,12,9,2,6? A. 12 B. 9 C. 54 D What is the median of following data 5, 7,10,3,8? A. 7.5 B. 7 C. 6 D If the number of books in a sample of five boxes are follow 11,8,2,2,7,7,2,5. Then the set is said to have A. Multimodal B. Unimodal C. Zero D. No mode 27

29 6-Find the midrange (MR) for the following data 7,-5,2,10,15 A. 2 B. 5 C. -5 D If the mean of 5 values equals 64, then A B. 320 C. 64 D. 200 X? 8-Find the mode for the following data A. 2,3 and 6 B. 2 C. No mode D. 3 and 6 9-The measures of central tendency for the following data 1,3,9,11,2 are: A. Mean=5.2 median=3 mode=no mode B. Mean=6.5 median=9 mode=zero C. Mean=2 median=3 mode=zero D. Mean=5.5 median=10 mode=no mode 10-The cost of four toys in a certain toy shop is given : $15,$20,$32,$1250 Which measure of central tendency should be used? A. Mode B. Mean C. Midrange D. Median 28

30 11-The is a measure of central tendency should be used when the data are qualitative. A. Median B. Range C. Mean D. Mode 12-What is the appropriate measure for the data that represents the marital status (married, divorced, widowed, single) A. Median B. Range C. Mean D. Mode 13-When the distribution is the positive skewed ;the relationship of mean,median and mode will be: A. Mean=Median=Mode B. Mean>Median>Mode C. Mean<Median<Mode D. The exact relationship cannot be determined 14-Find the mean and sample standard deviation for the following data set: 10, 5, 15, 20, 30, (Hint:x ). A) mean= 16, standard deviation= B) mean= 16, standard deviation= 9.62 C) mean= 16, standard deviation= 92.5 D) mean= 16, standard deviation= If a sample size is 9 and the standard deviation 7 then the variance is: A. 49 B. 2.6 C. 2 D

31 16-If the Cvar for English final examination was 6.9% and Cvar for History final examination was 4.9%. Compare The variations. A. The English class was more variable B. The History class was more variable C. Both of classes has the same variation D. Cannot determined 17-The mean of the number of sales of houses over a 3-month period is 56, and variance is 36, then the coefficient of variation is : A. 10.7% B % C. 64.3% D % 18-When a distribution is bell-shaped, approximately what percentage of data values will fall within 1, standard deviation of the mean? A. 95% B % C. 68% D. 99.7% 19-The mean of a distribution is 80 and the standard deviation is 7, if the distribution is normal, then approximately99.7% of the data will fall between A. 50 and 80 B. 59 and 101 C. 66 and 94 D. 73 and 87 30

32 20-The data value that is smaller than Q1 1.5 A) Minimum B) Median C) Quartile D) Outlier IQR 21-From the box plot below which class is more variable: Class A, is said to be: Class B A) Both class are the same B) Class A is more variable than class B C) Class B is more variable than class A D) Cannot be determined 22-Find the Z-score for the value75,when the mean is 80 and the standard deviation is 5 A. Z= B. Z= -1 C. Z= 1 D. Z= Which is not part a five-number summary? A. The mean B. The median C. The smallest and the largest data values D. 1 and 3 31

33 *Use the following boxplot graph to answer questions(24-26) IQR is approximately: A. 60 B. 0 C. 80 D The distribution shape is : A. Negative skewed B. Symmetrical C. Left skewed D. Positive skewed 26-The minimum value is A. 30 B. 10 C. 90 D

34 Question of chapter : Data Description 1. The largest value is possible to the non-outlier values for the data is shown in the next boxplot, a)13 b)12.5 c)13.5 D)12 2. The difference between a S 2 and a σ 2 is due to a)standard error of the mean c)sampling error b)sampling distribution D) no answer 3. if the majority of the data values fall to the right of the mean, then the distribution is skewed. a)negatively b)symmetric c) no answer D)positively 4. The mean of the waiting time in an emergency room is 78.2 minutes with a standard deviation of 13 minutes for people who are admitted for additional treatment. The mean of the waiting time for patients who are discharged after receiving treatment is minutes with a standard deviation of 18.6 minutes. Which waiting time are more variable? a)we cannot determine b) the two waiting time have same variable c)the waiting time patients who are discharged after receiving treatment are more variable D)the waiting time in an emergency more variable 33

35 5. is an extremely high or an extremely low data value when compared with the rest of the data values. a)percentiles b) an outliers c)there is no answer D) A Z score 6. Using the grades X 1, X 2, X 3, X 4, X 5, X 6, X 7, where X 1 < X 2 < X 3 < X 4 < X 5 < X 6 < X 7 to find the percentile rank of a grade of X 4 a)53th b)50 th c)52th D)46th 7. Given this boxplot The maximum value is a)10 b)12 c)13 D)11 8. The value corresponding to the 25th percentile is a)6 b) 5 c)4 D)7 9. The average score on a state CDL license exam is 75 with a standard deviation of 6. Find the corresponding z score for score 63. a)-5 b)0 c) -2 D) -3 34

36 10. Which has a better relative position: a score of 66 on an English test with a mean of 56 and a standard deviation of 18 or a score of 33 on an Arabic test with a mean of 28 and standard deviation of 9? a)a score of English test is better than a score of Arabic test b) A score of Arabic test is better than score of English test c) the two score are the same D) we cannot determine , 44, 12, 26, 36, 7 The Q 2 = a)19 b) 26 c)22.5 D) , 18, 26, 37, 10, 47 The Q 1 = a)5 b)18 c)26 D) , 13, 18, 4, 29, 36 The Q 3 = a)36 b)46 c)18 D)29 35

37 14. 9, 30, 36, 14, 19, 48 The IQR = a)19 b)20 c)22 D )23 36

38 محمد عمران لالستفسار او زياره الموقع خصم خاص للمجموعات حاله في عدم الرد نرجو ترك رساله باالسم والمادة