Stat 110 رياضيات واحصاء السنة التحضيرية دوري ثاني

Transcription

1 Stat 110 دوري ثاني محمد عمران السنة التحضيرية رياضيات واحصاء

2 محمد عمران حماضرات اون الين للتسجيل او االستفسار زياره الموقع االتصال حاله في عدم الرد نرجو ترك رساله باالسم والمادة

3 stat 110 Ch 3 Measure of variation مقاييس 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 1

4 stat 110 Ch 3 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 فان المجتمع األول )العينة األولى( أكثر تغيرا )عيىه أخرى( 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? Sol / محمد عمران رياضيات واحصاء للمرحله الجامعيه 2

5 stat 110 Ch 3 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 Sol الترتيب / محمد عمران رياضيات واحصاء للمرحله الجامعيه 3

6 stat 110 Ch / محمد عمران رياضيات واحصاء للمرحله الجامعيه 4

7 stat 110 Ch 3 Percentile percential للحصول على p 1 -نرتب االعداد ترتيب 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 Sol التستيب 2,3,5,6,8,10,12,15,18,20 2) Find the percentile rank for a score 6 Sol / محمد عمران رياضيات واحصاء للمرحله الجامعيه 5

8 stat 110 Ch 3 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 Sol الىاتج عدد وسبى العدد الرى وسبته % 25 زتبته العدد الثالث= 5 25th 2)60th percentile Sol عدد صحيح 2,3,5,6,8,10,12,15,18, / محمد عمران رياضيات واحصاء للمرحله الجامعيه 6

9 stat 110 Ch 3 Z score or standard score Measures of position مقاييس 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 Sol X / محمد عمران رياضيات واحصاء للمرحله الجامعيه 7

10 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 8

11 stat 110 Ch 3 A box plot Exploratory data analysis استطالع وتحليل البياوات 1) The lowest value of the data 2) 3) median 4) 5) the heights value of the data وتسمي هره القيم مجموعه البياوات 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 33,38,43,30,29,40,51,27,23,31,42 Sol Q 1 Q 3 Min Q 2 Max / محمد عمران رياضيات واحصاء للمرحله الجامعيه 9

12 stat 110 Ch 3 Find Q, Q, Q for the data set ,13,6,5,12,50,22,18 Sol Q Find Q, Q, Q for the data set , 18, 22,19,3,21,17,20 Sol Q / محمد عمران رياضيات واحصاء للمرحله الجامعيه 10

13 stat 110 Ch 3 Outliers 1) Find Q 1 Q 2 Q 3 2) find I QR Q 3 Q 1 3) Find Q IQR Q IQR 4) The low value of Q I QR and height value of Q IQR called outliers Find outliers for data set 24,32,54,31,16,18,19,14,17,20 Sol Example Q / محمد عمران رياضيات واحصاء للمرحله الجامعيه 11

14 stat 110 Ch 3 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 Sol / محمد عمران رياضيات واحصاء للمرحله الجامعيه 12

15 stat 110 Ch 3 Note box The distribution is approximately symmetric Symmetric 1( إذا كان الوسيط قسيبا مه مسكز إل كان الوسيط إلي يميه مسكز إل box The distribution is negatively skewed (left skewed) 2( إذا Left skewed كان الوسيط إلي يساز مسكز إل box The distribution is positively skewed (right skewed) 3( إذا Right skewed / محمد عمران رياضيات واحصاء للمرحله الجامعيه 13

16 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 14

17 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 15

18 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 16

19 stat 110 Ch 3 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 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

20 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 18

21 stat 110 Ch 3 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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 19

22 stat 110 Ch 3 *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 / محمد عمران رياضيات واحصاء للمرحله الجامعيه 20

23 stat 110 Ch 4 Correlation االرتباط scatter plots شكل اسكتور Is a statistical method to determine whether a relationship between variables is visual way to describe the nature of the relationship between the independent and dependent variables Type of correlation 1 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

24 stat 110 Ch 4 The correlation coefficient Is measures of how variables are related, it measures the strength and direction of a linear relationship between two variables The correlation coefficient.the symbol for the sample correlation coefficient is (rho) The range of the correlation coefficient is[ ] 1-if r=-1: there is a perfect (complete) negative linear relationship. 2-if r=1: there is a perfect (complete) positive linear relationship. 3-if r=0: there is no linear relationship (does not exist) The Pearson correlation coefficient معامل ارتباط بيرسون يستخدم مع البيانات الكميه r = n xy x y (n x 2 ( x 2 (n y 2 y 2 2 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

25 stat 110 Ch 4 نوع العالقة مدى العالقة = Prefect (complete )positive Strong positive Moderate positive Weak positive = No relation Weak negative Moderate negative Strong negative = Prefect (complete) negative 3 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

26 stat 110 Ch 4 Notes 1-apositive relationship exists when both variables increase or decrease at the same time 2-anegative relationship exists when one variable increases and the other variable decreases 3-if there a strong positive linear relationship between the variables, the value of r will close to +1 4-If there is a strong negative linear relationship between the variables, the value of r will be close to اذا كانث العالقة طرديه فكلما زادت Xزادت Y وكلما نقصث Xنقصث Y X increasing Y increasing X decreasing Y decreasing 2- اذا كانث العالفة عكسيه فكلما زادت Xنقصث Y وكلما نقصث Xزادت Y X increasing Y decreasing X decreasing y increasing 4 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

27 stat 110 Ch 4 Example Use the following to answer questions In the study of relationship between the number of absences X and the grade Y Of 8 student in the statistic class, the data are shown as follows = = = = = 1)The value of the Pearson correlation coefficient means that there is a...linear relationship between the number of absences and the final grade A) Strong negative b) strong positive c) moderate negative d) moderate positive Sol r = =.59 2)The range of the Pearson correlation coefficient value (r) for the positive linear relationship is A)0 B) C) D) 3)If the value of the correlation coefficient equal (-0.95),then the type of relationship is. A)weak negative B)strong negative C)strong positive D)weak positive 4)If the value of the correlation coefficient equals (-0.19), then the type of relationship is.. A)weak negative C)strong positive / B)strong negative D)weak positive 5 محمد عمران رياضيات واحصاء للمرحله الجامعيه

28 stat 110 Ch 4 5)If Pearson correlation coefficient ( equals (.045), then the relationship can be described as A)positive,strong B)positive,moderate and non linear C) weak and linear D) positive, moderate and linear 6)The correlation coefficient between the amount fats which person eats and his or her weight may be A) close to 1 B) close to 2 C) close to -1 D)0 7)If the correlation between body weight and annual income were high and positive, we could conclude that high income people A)are eating more food B)are losing weight C)are eating less food يكتسب وزن D)are gaining weight 8)A negative relationship between two variables means that for the most,as the X variable decreases,the Y variable A)stay the same B)increases c)decreases D)equals X 6 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

29 stat 110 Ch 4 The spearman rank correlation coefficient ) معامل الرتب يستخدم مع البيانات الوصفية )النوعية = ( When: n is the number of data pairs d is difference in ranks dهو فرق الرتب 1-The rang of Notes =[-1,1]or -1 r_s 1 2- if D=0 if both sets of data have the same rang =1 3-If there is no relationship between the two variables =0 7 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

30 stat 110 Ch 4 Example Two students were asked to rate six different television shows on a scale from 0 to 10 points The data are shown in the following table Show A B C D E F Student Student What is the Spearman Rank Correlation Coefficient for this set of data A) B) C)0.2 D) Solution d 2 = r = 1 6(6 2 1 = محمد عمران رياضيات واحصاء للمرحله الجامعيه /

31 stat 110 Ch 4 Use the table to find spearman rank correlation coefficient Home work problem Exam score Solution Home work Exam score = = ( = 9 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

32 stat 110 Ch 4 Example X Y Find the spearman rank correlation coefficient Solution X Y = = ( 2 = 11 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

33 stat 110 Ch 4 Use the following to answer questions If the differences between the rank of two variables are( then answer the following questions 1)The sample size is A)0 B)30 C)16 D)8 2) The value of the correlation coefficient is A)0.357 B) C) D)0.643 Solution = = = ( = 3)The correlation coefficient value means that there is A)very weak linear relationship B)moderate linear relationship C)very strong linear relationship D)no linear relationship 4)If the spearman rank correlation coefficient( equals 0.6, then the relationship can be described as A)positive, strong and non linear B)weak and linear C)positive,moderate and linear D)moderate and non linear 11 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

34 stat 110 Ch 4 Use the following to answer question X Y )The sample size is.. A)4 B)8 C)0 D)18 2)The spearman rank correlation coefficient ( is.. A)1 B)0 C)-1 D)-0.98 Solution X Y d = = ( = 3)The variable off the spearman rank correlation coefficient( means that there is. A)strong negative linear relationship B)moderate linear relationship C)strong positive linear relationship D)no linear relationship 12 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

35 stat 110 Ch 4 Regression االنحدار Regression is a statistical method used to describe the nature of the relationship between variable in a simple relationship,there are only two type of variables under,an independent variable (explanatory variable)or predictor variable and a dependent variable(outcome variable)or response variable Equation of regression line is = = Where b is the slope of the line And a is y-intercept 2 ( 2 = = 2 2 ( Or 2 13 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

36 stat 110 Ch 4 Round the values of a and b to three decimal places another name is line of best fit Remark 1-The regression line can be used to predict a value for the dependent variable (Y) for a given value of the independent variable(x) 2- اشاره معامل االرجباط r هي نفس اشاره امليل b b اذا كانث العالفة طرديه فانه كلما زادات مبقدار 1 زادت اذا كانث العالفة عكسيه فكلما زادات y مبقدار مبقدار 1 قلث y مبقدار x x b محمد عمران رياضيات واحصاء للمرحله الجامعيه /

37 stat 110 Ch 4 Example Instructor is interested in finding the relationship between the final exam grade of student enrolled in calculus Ι (x) And calculus I (y) Use the following information to answer the following question = = = = = = Slope(b)= the Pearson correlation coefficient is : A)0.94 r = B)0.81 Sol C)0.68 D) = 0 94 [ ] [ ] 2-interpret the value of the person correlation coefficient (r) A)weak positive linear correlation B)weak negative linear correlation C)strong positive liner correlation D)strong negative linear correlation 3-the equation of the regression line is A) = X B) = X C) = X D) = X 4-predict a calculus Πexam score for student who get 80 in calculus 1 A)85 B)80 C)70 D)76 y= = 15 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

38 stat 110 Ch 4 Age and sick days Age x Days Y find when X=47years Example solution = = = = = = = = ( = = Eq = = = The slope of regression line =-2x +5 is : A)2 B)5 C)-5 D)-2 Suppose that every man's age (x) is exactly twice his wife's age (y).then the Pearson correlation coefficient (r )between x and y is A)0.5 B)-0.5 C)1 D)-1 16 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

39 stat 110 Ch 4 Use the following to answer question The equation of the regression line between persons age in year (x)and the number of hours he exercises per weak (y) is given by = X 1) The correct statement that represent the relation between x and y A) when the number of hours he exercises increases by 1 hour, his age increases by on average B)when the number of hours he exercises decreasesby1 hour, his age decreases by on average C)when persons age increases by 1year, the number of hours he exercises decreasesby0.18on average D) When persons age increases by 1 year, the number of hours he exercises increases by 0.18 on average 2) Predict the number of hours a person exercises per week when his age is 50 year A)1.49 B)10.04 C)19.9 sol D)1.944 = = There is an approximate linear relationship between the height(y in cm) and (x from 5 to18years) describe by =50+6X Which of the following is not correct A)children reach the height of 50cm when they are 50/6=8.33years old B)children's heights increase by 6cm for each year they grow older C)the estimated height of a 10years old child is 110 D) my son is 7 years old and is 110 cm tall. He is taller than average 17 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

40 stat 110 Ch 4 When the correlation (r) equals zero,the linear relationship between the variables.. A)is strong B)is weak C)is moderate Use the following to answer the question D)does not exist the following data is recorded to determine the relationship between the number of hour a person goes without sleeping (x) and number of mistakes he makes on a simple test (y) N=10 = = = = = 1-the Pearson correlation coefficient (r )is A)0.24 B)0.33 C)-0.24 D)-0.33 r = sol = 0 33 ( ( )The value of the Pearson correlation coefficient (r )means there is. A)strong negative linear relationship C)strong positive linear relationship 3)the equation of the regression is = = = = Sol نوجد قيمه b B)moderate negative linear D)weak positive linear = = 18 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

41 stat 110 Ch 4 4)The correct statement that represents the relationship between (x) and (y) A)when the number of hour a person goes without sleeping increases by1hour,his number of mistake increasesby1.02 on average B)when the number of hours a person goes without sleeping increases by 1 hour,his number of mistake decreases by 1.02 on average C)when the number of mistakes increases by 1,the number of hours a person goes without sleeping decreases by 1.3 on average D)when the number of mistakes increases by 1 year, the number of hours a person goes without sleeping increases by 1.3 on average 5)Predict the number of mistakes for a person who goes 9.51 hours without sleeping A)-8 B)11 sol C)13 D)-11 = = 6) If the equation of regression line between X & Y is y^ = -2x + 4 The relationship between x and y is a) No linear B) no answer c) Positive linear D)Negative linear 19 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

42 stat 110 Ch 4 Use the following to answer question In the study of relationship the number of absences X and the final grade Y of 6 student is statistic class, the data are = = = = = 1)The correlation coefficient is A)0.82 B)0.92 C)-0.82 sol D)1 = ( ( = 2)The slope of regression line is A)2.45 B)-3.45 C)3.45 sol = = D) )A negative relationship between two variables means that for the most part, as the X variable,the Y variable A)decreases B)increases C)equals X D)remains the same 21 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

43 stat 110 Ch 4 4)In the relationship between the number of studying hours and an exam grade, the number of studying is assumed to be A)continuous variable C)nominal variable B)dependent variable D)independent variable 5)The variable that is used to predict the value of another is called A)independent variable B)dependent variable C)correlation The variable that is used to predict the value of another is called A)response variable B)explanatory variable C)correlation variable D)variable of determination 21 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

44 stat 110 Ch 4 A home owner wanted to determine if there was a relationship between the size (x in 100square feet) of a new home and the price y (in thousands of dollars)of the home he found the equation of the regression line and graph the line in a scatter plot as presented below = answer the following four question 1)What would be the predicted home price if = A)461.4 B) C)350 D) sol = = 2)When the size of home increases one unit(100square feet),the expected change in home price is A) B) C)0 D) )the size of the home called A)explanatory variable C)response variable B)outcome variable D)confounding variable 4)The line of best fit means the line that of the.distance from each point to this line is at a minimum A)squares of the vertical C)squares of the horizontal B)vertical D)horizontal 22 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

45 stat 110 Ch 4 in the =, what is A)slope of the regression line. B)intercept of the regression line C)predicted value of given a specific D)value of y when x=0 value. Compute the value of the Pearson product moment correlation coefficient for the data below : A) x value y value B)0.882 C)0.924 SOL = = = = = D) = ( ( ( = 23 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

46 stat 110 Ch 4 Complete the following statement s with the be answer. 1-the regression line is called the 2-if all the point fall on a straight line the value of..will be..or.. 3-the strength and direction of the relationship between two variable is determined by the value of the 4-the equation of the regression line used in statistics is. 5-the sign of r and.. will always be the same 6-a statistics graph of two variable is called 7-the rang of the correlation coefficient is from.to 1)The Range of Correlation Coefficient is A)-1 to 1 B)0 to 1 C)-2 to 1 D)Fisrt and Second Options 2)Which of the following values represent a correlation coefficient A)-1.0 B)1.1 C)2.2 D)all of the above 3)Determine the relatioship between the vaiables if the correlation coefficient = 0.2 A)No linear relationships B)weak positive C)strong positive 4)If 'y' is the dependent variable and 'a' is the y-intercept and 'b' is the slope then the equation of a regression line is: A) a bx B) x a b C)x b ax D)Fisrt and Second Options 24 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

47 stat 110 Ch 4 5)In the equation of regression line y = a + bx, 'b' is called A)Slope of the regression line B)Marginal change C)Y-intercept D)Fisrt and Second Options 6)If the equation of regression line y = x. What is the value of y when x = 20 A) 0.68 B) 6.68 C) 3.68 D) )What is the relationship between the sign of the correlation coefficient and the slope of the regression line? A)When r is negative, b D)First and will be negative Second Options B)When r is positive, b will be positive C)When r is positive, b will be negative 8)If the points in a scatter plot are moving up then the relation between the variables is A)Positive B)Negative C)No linear relationships D)all of the above 9)If all the points on a scatter plot fall on a straight line going up the value of r will be A) +1 B) -1 C) +2 D) none of the above 10)If the differences between the ranks of two variables are (1,-1,1,-1) The sample size is )The equation of the regression line between an age of person in years (x) and the number of hours he exercises per week (y) is given by y = x Predict the number of hours a person exercises per week when his age is 49 years. A)26.6 B)28.6 C)25.6 D) محمد عمران رياضيات واحصاء للمرحله الجامعيه /

48 محمد عمران حماضرات اون الين للتسجيل او االستفسار زياره الموقع االتصال حاله في عدم الرد نرجو ترك رساله باالسم والمادة

49 stat 110 Ch 5 Probability and counting rules االحتماالت ووظريه العد Probability as general concept can be defined as the chance of an event occurring *probability is used in game of chance, investment, weather, forecasting and in various other area Probability experiment An out come Sample space An event Simple event Compound event Equally likely events Ven diagram is chance process that lead to well defined called out come Is the result of single trial of probability experime Is the set of all possible outcomes of experiment Consists of asset of outcomes of probability experiment An event with one out comes An event with more than one out come Are events that have the same probability of occurring Are used to represent probability pictorially 1 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

50 stat 110 Ch 5 Example Find the sample space for 1-toss on coin * + 2-toss two coins *( ) ( ) ( ) ( )+ 3-roll the dice * + 4-roll two dice { } 2 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

51 stat 110 Ch 5 Note Tree diagram. is a device used to list a sequence of events is systematic way Example Find the sample space for the gender of the children if a family has three children?and give an example for a sample event and another one for compound event. Use B for boy and G for girl b g g b b b g Solution b g b g g g b s bbb bbg bgb bgg gbb gbg ggb ggg Sample event =*bbb+ Compound event = bbg gbb bgg 3 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

52 stat 110 Ch 5 1)Classical probability Type of probability The probability of any event E is ( ) ( ) ( ) ( ) 2)Empirical probability Based on observation ( ) ( ) 3)Subjective probability Based on estimate and in exact information classical empirical Subjective يبنى على التخمين -نسبة مئىية كسىر ويشير إلى المستقبل - فبصلة عشرية next... 4 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

53 stat 110 Ch 5 Example Classify each statement as in example of classical probability, empirical probability or subjective probability A)the probability that a person will watch news in the 6 o'clock evening is 0.15 empirical B)the probability of winning at a chuck-a-luck game is C)the probability that abuse will in an accident on a specific run is about 6% empirical D)the probability of getting a royal flush when five cards are selected at random is Classical Classical E)the probability that an new fast-food restaurant will be a success in Chicago is 35% empirical F)the probability that student will get a C or better in a statistics Course is about 70% empirical G)the probability that interest rates will rise in the next 6 month is 0.50 subjective 5 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

54 stat 110 Ch 5 Probability ( ) *the rang of the value of the probability =, - * ( ) where E can never occur * ( ) where E certain occur * ( ) in the samples pace the sum of the probabilities of all out comes equal 1 Example A probability experiment is conducted.which of these cannot be consider a probability of outcomes? Answer 6 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

55 stat 110 Ch 5 If Adie is rolled one time, find these probability? A)of getting a 4 ( ) B)of getting an even number Even =* + ( ) C)of getting a number greater than 4 Greater than 4=* + ( ) D)of getting a number less than 7 ( ) E)of getting a number greater than 0 ( ) F)of getting a number greater than 3 4,5,6 ( ) G)of getting a number greater than 3 or an odd number Greater than 3=4,5,6 or odd=1,3,5 Greater than 3 or odd=1,3,4,5,6 H)of getting a number greater than 3 and an odd number Greater than 3 and odd =5 7 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

56 stat 110 Ch 5 Example A couple has three children, find each probability. A)all boys B)all girls C)exactly two boys or two girls D)at least on child of each gender Solution * + A- * + b- * + c -* + D- * + 8 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

57 stat 110 Ch 5 If two dice are rolled one time,find the probability of getting these results A)assume of 6 B)doubles C) assume of 7 or 11 D)assume greater than 9 E)assume less than or equal to 4 Solution a)* + b)* + c)assume 7=* + assum11=* + assume 7 or 11=* + d)* + e)* + 9 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

58 stat 110 Ch 5 Example Rolled two dice and multiply the number to gather. find? A)write out the sample space. B)what the probability that the product is a multiple of 6? C)what is the probability that the product is less than 10? Solution Sample space 1 ضرب b) c) 11 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

59 stat 110 Ch 5 Example For a recent year,51% of the families in the united states had no children under the age of 18,20%had one children,19%had two children,7%had three children,and 3% had four or more children. If a family is selected at random,find the probability that the family has A)two or three children. B)more than one child. C)lees than three children. D)based on the answer to part a, b and c, which is most likely of occur?explain why? Solution a) =0.26 b) =0.29 c) =0.80 d)the most likely is C is more than a and b 11 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

60 stat 110 The complementary events Ch 5 مكمله الحىادث The complement of an event E is the set of out comes in the sample space that are not included in the outcomes of event E The complementary event are a mutually exclusive. Find a complement of each event A)rolling Adie and getting 4 ( ) ( ) Rules ( ) ( ) ( ) ( ) ( ) ( ) B)selecting a letter of alphabet getting avowal. ( ) ( ) C)selecting a month and getting a month that begging with a j D) Selecting a day of the weak and getting a weak day if the probability that person lives in an industrialized country of the world is.find probability that person does not live in an industrialized country sol 12 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

61 stat 110 Ch 5 1)An event and its complement are A) Mutually exclusive B)independent C) not mutually exclusive D)dependent 2)Two event are if they cannot occur at the same time A)not mutually exclusive B)dependent C)mutually exclusive C)independent 3)Two event that can occur at the same time are called A)mutually exclusive B)dependent C)not mutually exclusive D )independent 4)Two event are said to be.. if the probability of one event occurring is affected by the event A)not mutually exclusive B)independent C)mutually exclusive C)dependent 5).. probability uses a frequency distribution to compute probabilities A)classical B)empirical C)subjective D)continuous 6)Student and a professor each choose a number between 1 and 5 (1 and 5 are both possible choice). What is the probability that the two choose the same odd number? A)0.04 B)0.03 C)0.12 D)0.16 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) {( ) ( ) ( ) ( ) ( )} *( ) ( ) ( )+ 13 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

62 stat 110 Ch 5 7)Consider the experiment of selecting one item at random from a box containing an equal number of defective (D) and non-defective (N) items.the sample space for this experiment is A)2 B) * + C)4 D) * + 8)If a die is rolled, the event of getting an eve number would be called A)compound event B)simple event C)sample space 9)If there is 20% chance that it will rain tomorrow.what is the probability that it will not rain tomorrow A).08 B)0.20 C)0.80 D) = )If there are 20equally likely events, then the probability of the second one occurring is.. A)1 B)1/10 C)1/20 D)0 P=1/20 11)the complement of guessing 10 incorrect answer on 10 question true/false exam is A)guessing 10 incorrect answer B)guessing at least 1 incorrect C)guessing at least 1 correct answer C)guessing 10correct answers 12)The complement of guessing at least 5 correct answer an a 6 question true/false exam is.. A)guessing 6 incorrect answer B)guessing at least 2 incorrect answer C)guessing at least 2 correct answer D)guessing 6 correct answer 13)The complement of guessing 4 correct answer a 4 question true/false exam is A)guessing 4 incorrect answers B)guessing at least 1 incorrect C)guessing at least 1 correct answer C)guessing no incorrect answer 14 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

63 stat 110 Ch 5 14)When 10 adults are tested for high blood pressure, the complement of at least one of the results are positive A)none of the adults have high blood pressure B)nine of the adults have high blood pressure C)all of the adults have high blood pressure D)one of the adults have high blood pressure 15)A card is drown from ordinary dice answer the following 1. What is the probability of getting a 5. (A) 5/52 (B) 2/26 (C) 3/26 (D) 5/13 2. What is the probability of getting a 10 or a heart? (A) 3/13 (B) 5/52 (C) 2/13 (D) 4/13 3. What is the probability of getting a picture card? (A) 3/13 (B) 5/26 (C) 1/13 (D) 6/13 4-What is the probability of getting a 6 and Black? (A) 2/52 (B) 2/26 (C) 1/26 (D) 5/13 15 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

64 stat 110 Ch 5 Consider the table to answer the following two question Cook type Number sold Chocolate chip 20 Peanut butter 15 Oat meal 30 sugar 10 1)What is the level measurement for the cook type A)discrete B)ordinal C)nominal D)continuous 2)What is the probability of selecting a chocolate chip cook A)4/15 B)1/2 C)1/15 D)1/4 Use the following to answer the question Two dice are rolled let X represent the summation of the two face that will appear 1-the probability of A)0.167 B)0 C)0.028 D) the probability of A)0 B)0.083 C)0.056 D)1 ) *( ) ( ) ( )( )( ) ( )+ ) كل االعداد مجمىعها اقل مه او تساوي محمد عمران رياضيات واحصاء للمرحله الجامعيه /

65 stat 110 Ch 5 P=1 Use the following to answer the question If two dice are rolled and x is the random variable represent the summation of two face that will appear 1-the probability of X =4 A)0.083 B)0.139 C)0.861 D) the probability of X=12 A)0.972 B)0.917 C)0.028 C)0 3-the probability of X=15 A)0.056 B)0.028 C)0.083 D)0 ) *( )( ) ( )+ ) *( )+ ) 17 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

66 stat 110 Ch 5 the edition rules *when two event A and B are mutually exclusive, the probability that A or B will occur is ( ) ( ) ( ) *when two event A and B are not mutually exclusive, the probability that A or B will occur ( ) ( ) ( ) ( Determine whether these events are mutually exclusive. A) roll die: get an even number, and get a number less than 3 Not mutually exclusive B) roll die: get a prime number,and get an odd number Not mutually exclusive C) roll die: get a number greater than 3, and get a number less than 3 Mutually exclusive D) select student in your class.the student has blond hair, and the student has blue eyes Mutually exclusive E) select a student in your collage.the student is sophomore, and the student is a business major Mutually exclusive F) select any course. It is a calculus course, and it is an English course Mutually exclusive E) select a registered voter, the voter is a republican, and the voter is a democrat Mutually exclusive 18 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

67 stat 110 Ch 5 Example 1)A box contains 3 glazed doughnuts, 4 jelly doughnuts and 5 chocolate doughnuts. Find the probability that it is either a glazed doughnut or chocolate doughnuts? 2)a day of the week is selected at random. Find the probability that it is a week day (Thursday or Friday) 3)An urn contains 6 red balls, 2 green balls,1 blue ball and 1 white ball. If a ball is drawn find the probability of getting A)red or green C)not green B)blue or white D)blue and white ) ( ) ) ( ) ) ( ) ) ( ) 19 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

68 stat 110 Ch 5 4)The probability that a student has a computer is 0.82 and the probability he has a car is 0.48 while he probability that he has either a computer or a car is find the probability that the student has both A)0.38 B)0.44 C)0.34 D)0.52 ( ) ( ) ( ) 5)The probability that student has a computer is 0.91 and the probability he has a car is 0.49 while the probability that he has both is find the probability that the student has a computer or he has a car. A)0.82 B)0.51 C)0.94 D)0.65 ( ) ( ) ( ) ( ) 21 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

69 stat 110 Ch 5 6)The bargain auto mall has these cars in stock Suv compact Mid-size Total Foreign Domestic total If a car selected at random, find the probability that it is A)Domestics B)Foreign and mid-size C)Domestic or an suv ) ( ) ) ( ) ) ( ) ( ) ( ) ( ) 21 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

70 stat 110 Ch 5 7)A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown Gender Yes No Total Male Female Total Find the probabilities. A)the respondent answered yes and that the respondent was a female. B)the respondent was a male or that the respondent answered no ) ( ) ) ( ) ( ) ( ) ( ) 22 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

71 stat 110 Ch 5 8)Use the following to answer the question An apartment building has the following apartments. 1 st floor 2 nd floor If an apartment is selected at random, 3 rd floor Total 2 bedroom bedroom Total (1)what is the probability that it is on the 2 nd floor or 3 bedroom A)4/12 B)6/12 C)7/12 D)3/12 ( ) ( ) ( ) ( ) (2)what is the probability that it is 3 bedroom apartment or it is the 3 rd floor A)2/12 B)6/12 C)8/12 D)1/12 ( ) ( ) ( ) ( ) 23 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

72 stat 110 Ch 5 9)In a recent study, the following data was obtained in response to the question "do you favor recycling in your neighborhood" Yes No No opinion Total Males Female Total If a person is selected at random, use the above table to answer the following question 1-the probability that a person is a female or she answered yes regarding recycling is A)0.67 B)0.32 C)0.767 D)0.233 ( ) ( ) ( ) ( ) 2-what is the probability that a person has no opinion regarding recycling? A)0.267 B)0.333 C)0.4 D) what is the probability that a person is male and the answered no regarding recycling? A)0.107 B)0.227 C)0.093 D) the probability that a person is a male or he has no opinion regarding recycling is A)0.467 B)0.8 C)0.747 D)0.533 ( ) ( ) ( ) ( ) 24 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

73 stat 110 Ch 5 10)The manger of a bank recorded the amount of time each customer spent waiting in line during peak business hours one Monday.the frequency table below summarizes the results. Waiting time (minutes) Total 52 Number of Customers If we randomly select one of the customers represented in the table, what is the probability that the waiting time at least 12 minutes or between 8 and 15 minutes? A)0.519 B)0.63 C)0.558 D)0.2 ( ) ( ) ( ) ( ) 25 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

74 stat 110 Ch 5 11)Use the following to answer questions. A supermarket employs cashiers, managers and cleaner. the distribution of employs according to mantel status is shown here Find the probability that 1-the employee is a manger or married Cashier Manager Cleaner Total Married Not married Total A)24/46 B)12/46 C)27/46 D)39/46 ( ) ( ) ( ) ( ) 2-the employee is a casher or he is married A)9/24 B)9/22 C)29/46 D)9/14 ( ) ( ) ( ) ( ) 26 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

75 stat 110 Ch 5 Example 1)In a hospital unit there are 8 nurses and 5 physicians, 7nurses and 3 physicians are females. If staff person is selected, find the probability that the subject is A)nurse or male B)physicians or female Nurse physicians Totat Female Male Total Sol ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) 27 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

76 stat 110 Ch 5 2)At a particular school with 200 male student, 58 play football, 40 play basketball,8 play both. What is the probability that a randomly selected male student A)play neither sport B)plays (football and basketball) C)plays (football or basketball). 200 Basket ball 32 8 Football 50 ) ( ) ) ( ) ) ( ) ( ) ( ) ( ) 28 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

77 stat 110 The multiplication rules Ch 5 Two event A and B are independent events if : A occur does not affect the probability of B occurring Multiplication rules 1 When two events are independent, the probability of both occurring is. ( ) ( ) ( ) Example 1)An urn contains 3 red balls,2blue balls, and 5 white balls. A ball is selected and its color noted. Then it is replaced. a second Ball is selected and its color noted. find the probability of each of these A)selecting 2 blue balls. B) selecting 1 blue ball and then 1 white ball. C)selecting 1 red ball and then 1 blue ball. ) ( ) ) ( ) ) ( ) 29 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

78 stat 110 Ch 5 2)Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If 3 men are selected at random, find the probability that all of them will have this type of red-green color blindness. ( ) 3)A coin is flipped and a die is rolled.find the probability of getting ahead on the coin and a 4 on the die ( ) 4)If 28%of U.S. medical degrees are conferred to women, find the probability that 3 randomly selected medical school graduates are men? ( ) ( ) 31 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

79 stat 110 Ch 5 5)Eighty-eight percent of U.S. children are covered by some type of health insurance.if 4 children are selected at random, what is the probability that none are covered? ( ) ( ) 6)A person owns a collection of 30 CDS,of which 5 are country music. If 2 CDS are selected at random, find the probability that both are country music. ( ) ( ) 7)in a civic organization, there are 38 members,15 are men and 23 are woman. If 3 members are selected to plan the July 4th parade, find the probability that all 3 are women ( ) ( ) 31 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

80 stat 110 Ch 5 Counting rules قبعده العد Fundamental Counting rule: In a sequence of n events in which the first one has k1 possibilities and the second event has k2 and the third has k3, and so forth, the total number of possibilities of the sequence will be K1 K2 K3 Kn Example 1)A coin is tossed and a die is rolled. Find the number of outcomes for the sequence of events. 2)The digits 0,1,2,3and 4 are to use in a four-digit ID card.how many different cards are possible if repetition are permitted? 3)The digits 0,1,2,3,4,5,6,7,8,and 9 are to be used in a four- digit ID card. how many different cards are possible if repetitions are permitted? 32 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

81 stat 110 Ch 5 4)ID cards consist of 5 letters followed by 2 digits. How many cards can be made if repetitions are allowed? 5)ID cards consist of 5 letters followed by 2 digits. How many cards can be made if repetitions are not allowed? 6)A paint manufacturer wishes to manufacture several different paint the categories include Color Type Texture Use red, blue,white, black, green, brown,yellow latex,oil flat,semi-gloss,high glass out door, in door How many different kinds of paint can be made if a person can select one color, one type, one texture and one use? 7)There are 2 major roads from city X to city Y and 4 major roads from a city Y to city Z. how many different trips can be made from city X to city Z passing through city Y? 33 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

82 stat 110 Ch 5 Permutations A permutations is an arrangement of N objects in a specific order. Factorial formulas. For any counting N ( )( ) Permutation rules The arrangement of N objects in a specific order using R objects at a time is called a permutation of N taking R objects at a time.it is written as And formula is الترتيب هاااام ( ) * order is important ( ) ( ) 34 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

83 stat 110 Ch 5 Example 1)How many different ways can a chairperson and an assistant. chairperson be selected for a research project if there are seven scientists available? الترتيب مهم 2)A store manager wishes to display 8 different brands of shampoo in arrow. How many ways can this be done? الترتيب مهم 3)A television news director wishes to use three news stories on an evening show. One story will be the lead story, one will be the second story and the last will be a closing story. If the director has a total of eight stories to choose from, how many possible ways can the program be set up? الترتيب مهم 35 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

84 stat 110 Ch 5 Combination rule The number of combination of R objects selected from N objects is denoted by and is given by the formula ( ) التريب غير مهم Example 1)How many combination of 4 objects are there, taken 2 at a time? الترتيب غير مهم 2)in a club there are 7 women and 5 men. a committee of 3 women and 2 men is be chosen. How many different possibilities are there الترتيب غير مهم 3)How many different test, can be made from a test bank of 20 questions if the test consists of 5 question? الترتيب غير مهم 4)How many ways can a dinner patron select 3 appetizers and 2 vegetables if there are 6 appetizers and 5 vegetables on the menu? الترتيب غير مهم 36 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

85 stat 110 Ch 5 5)How many different ways can an instructor select 2 text books from a possible 17? الترتيب غير مهم 6)A combination lock consist of the 26 letter of the alphabet if a 3 letter combination is needed combination will consist how many committee? الترتيب غير مهم 7)How many can selecting 2 mathematic books and 3 physics books from a box containing 4 mathematics books and 8 physics books? الترتيب غير مهم 8)There are 2 math-students and 5 stat-students in class how many ways can a group of 3students be selected if there must be at least one math-student on this group 37 محمد عمران رياضيات واحصاء للمرحله الجامعيه /

86 stat 110 Ch 5 Example 1)How many different 4-letter permutations can be formed from the letters in the word orange? A) 360 B) 840 C) 120 D) 420 الترتيب مهم 2) If the letters A,B,C,D,E, and F are to be used in a letter code consists of six digits, how many different codes are possible if the first letter must be A and repetitions are not permitted? A) 720 B) 1440 C) D) 120 نثبت ببقى الحروف 3)An ID card consists of 2 letters followed by 3 digits. How many different ID cards can be made if repetitions are allowed? A) B) C) D) ببلتكرار مسمىح 4)How many different 3-letter permutations can be formed from the letters in the word answer? A) 20 B)720 C) 6 D) 120 الترتيب مهم 5)If the 26 alphabetic letters and the numbers from 0 to 9 is to be used to form a car plate that consists of 3 letters and 4 numbers, how many different plates are possible if repetitions are not permitted? A) B) C) D) التكرار غير مسمىح 6)An ID card consists of 2 letters followed by 2 digits. How many different ID cards can be made if repetitions are allowed? A) B) C) D)67600 التكرار مسمىح 38 محمد عمران رياضيات واحصاء للمرحله الجامعيه /