Changing Unit of Measurement.ppt

1、week3,1,Changing Unit of Measurement,A linear transformation changes the original value x into a new variable xnew . xnew is given by an equation of the form,Example 1.21 on page 45 in IPS.(i) A distance x measured in km. can be expressed inmiles as follow, .(ii) A temperature x measured in degrees

2、Fahrenheit can beconverted to degrees Celsius by,week3,2,Effect of a Linear Transformation,Multiplying each observation in a data set by a number b multiplies both the measures of center (mean, median, and trimmed means) by b and the measures of spread (range, standard deviation and IQR) by |b| that

3、 is the absolute value of b.Adding the same number a to each observation in a data set adds a to measures of center, quartiles and percentiles but does not change the measures of spread.Linear transformations do NOT change the overall shape of a distribution.,week3,3,week3,4,Example 1,A sample of 20

4、 employees of a company was taken and their salaries were recorded. Suppose each employee receives a $300 raise in the salary for the next year. State whether the following statements are true or false. The IQR of the salaries will be unchanged increase by $300 be multiplied by $300 The mean of the

5、salaries will be unchanged increase by $300 be multiplied by $300,week3,5,Density curves,Using software, clever algorithms can describe a distribution in a way that is not feasible by hand, by fitting a smooth curve to the data in addition to or instead of a histogram. The curves used are called den

6、sity curves.It is easier to work with a smooth curve, because histogram depends on the choice of classes.Density CurveDensity curve is a curve that is always on or above the horizontal axis. has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution.,week3,6,Th

7、e area under the curve and above any range of values is the relative frequency (proportion) of all observations that fall in that range of values.Example: The curve below shows the density curve for scores in an exam and the area of the shaded region is the proportion of students who scores between

8、60 and 80.,week3,7,Median and mean of Density Curve,The median of a distribution described by a density curve is the point that divides the area under the curve in half. A mode of a distribution described by a density curve is a peak point of the curve, the location where the curve is highest. Quart

9、iles of a distribution can be roughly located by dividing the area under the curve into quarters as accurately as possible by eye.,week3,8,Normal distributions,An important class of density curves are the symmetric unimodal bell-shaped curves known as normal curves. They describe normal distribution

10、s.All normal distributions have the same overall shape.The exact density curve for a particular normal distribution is specified by giving its mean (mu) and its standard deviation (sigma).The mean is located at the center of the symmetric curve and is the same as the median and the mode.Changing wit

11、hout changing moves the normal curve along the horizontal axis without changing its spread.,week3,9,The standard deviation controls the spread of a normal curve.,week3,10,There are other symmetric bell-shaped density curves that are not normal e.g. t distribution. Normal density function is mathemat

12、ical model of process producing data. If histogram with bars matching normal density curve, data is said to have a normal distribution.Notation: A normal distribution with mean and standard deviation is denoted by N(, ).,week3,11,The 68-95-99.7 rule,In the normal distribution with mean and standard

13、deviation , Approx. 68% of the observations fall within of the mean . Approx. 95% of the observations fall within 2 of the mean . Approx. 99.7% of the observations fall within 3 of the mean .,week3,12,Example 1.23 on p72 in IPS,The distribution of heights of women aged 18-24 is approximately N(64.5,

14、 2.5), that is ,normal with mean = 64.5 inches and standard deviation = 2.5 inches.The 68-95-99.7 rule says that the middle 95% (approx.) of women are between 64.5-5 to 64.5+5 inches tall.The other 5% have heights outside the range from 59.5 to 69.5 inches, and 2.5% of the women are taller than 69.5

15、 .Exercise:1) The middle 68% (approx.) of women are between _to _inches tall.2) _% of the women are taller than 66.75. 3) _% of the women are taller than 72.,week3,13,Standardizing and z-scores,If x is an observation from a distribution that has mean and standard deviation , the standardized value o

16、fx is given by A standardized value is often called a z-score.A z-score tells us how many standard deviations the original observation falls away from the mean of the distribution.Standardizing is a linear transformation that transform the data into the standard scale of z-scores. Therefore, standar

17、dizing does not change the shape of a distribution, but changes the value of the mean and stdev.,week3,14,Example 1.26 on p61 in IPS,The heights of women is approximately normal with mean = 64.5 inches and standard deviation = 2.5 inches. The standardized height isThe standardized value (z-score) of

18、 height 68 inches isor 1.4 std. dev. above the mean.A woman 60 inches tall has standardized heightor 1.8 std. dev. below the mean.,week3,15,The Standard Normal distribution,The standard normal distribution is the normal distribution N(0, 1) that is, the mean = 0 and the sdev = 1 . If a random variab

19、le X has normal distribution N(, ), then the standardized variable has the standard normal distribution.Areas under a normal curve represent proportion of observations from that normal distribution. There is no formula to calculate areas under a normal curve. Calculations use either software or a ta

20、ble of areas. The table and most software calculate one kind of area: cumulative proportions . A cumulative proportion is the proportion of observations in a distribution that fall at or below a given value and is also the area under the curve to the left of a given value.,week3,16,The standard norm

21、al tables,Table A gives cumulative proportions for the standard normal distribution. The table entry for each value z is the area under the curve to the left of z, the notation used is P( Z z). e.g. P( Z 1.4 ) = 0.9192,17,Standard Normal Distribution,The table shows area to left of z under standard

22、normal curve,week3,18,The standard normal tables - Example,What proportion of the observations of a N(0,1) distribution takes valuesa) less than z = 1.4 ?b) greater than z = 1.4 ?c) greater than z = -1.96 ?d) between z = 0.43 and z = 2.15 ?,week3,19,Properties of Normal distribution,If a random vari

23、able Z has a N(0,1) distribution then P(Z = z)=0. The area under the curve below any point is 0. The area between any two points a and b (a b) under the standard normal curve is given byP(a Z b) = P(Z b) P(Z a)As mentioned earlier, if a random variable X has a N(, ) distribution, then the standardiz

24、ed variable has a standard normal distribution and any calculations about Xcan be done using the following rules:,week3,20,P(X = k) = 0 for all k.The solution to the equation P(X k) = p isk = + zpWhere zp is the value z from the standard normal table that has area (and cumulative proportion) p below

25、 it, i.e. zp is the pth percentile of the standard normal distribution.,week3,21,Questions,1. The marks of STA221 students has N(65, 15) distribution. Find the proportion of students having marks (a) less then 50. (b) greater than 80.(c) between 50 and 80.2. Example 1.30 on page 65 in IPS:Scores on

26、SAT verbal test follow approximately the N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?3. The time it takes to complete a stat220 term test is normally distributed with mean 100 minutes and standard deviation 14 minutes. How m

27、uch time should be allowed if we wish to ensure that at least 9 out of 10 students (on average) can complete it? (final exam Dec. 2001),week3,22,General Motors of Canada has a deal: an oil filter and lube job in 25 minutes or the next one free. Suppose that you worked for GM and knew that the time n

28、eeded to provide these services was approximately normal with mean 15 minutes and std. dev. 2.5 minutes. How many minutes would you have recommended to put in the ad above if it was decided that about 5 free services for 100 customers was reasonable? 5. In a survey of patients of a rehabilitation ho

29、spital the mean length of stay in the hospital was 12 weeks with a std. dev. of 1 week. The distribution was approximately normal. Out of 100 patients how many would you expect to stay longer than 13 weeks? What is the percentile rank of a stay of 11.3 weeks? What percentage of patients would you ex

30、pect to be in longer than 12 weeks? What is the length of stay at the 90th percentile? What is the median length of stay?,week3,23,Normal quantile plots and their use,A histogram or stem plot can reveal distinctly nonnormal features of a distribution. If the stem-plot or histogram appears roughly sy

31、mmetric and unimodal, we use another graph, the normal quantile plot as a better way of judging the adequacy of a normal model. Any normal distribution produces a straight line on the plot. Use of normal quantile plots:If the points on a normal quantile plot lie close to a straight line, the plot in

32、dicates that the data are normal. Systematic deviations from a straight line indicate a nonnormal distribution. Outliers appear as points that are far away from the overall pattern of the plot.,week3,24,Histogram, the nscores plot and the normal quantile plot for data generated from a normal distrib

33、ution (N(500, 20).,week3,25,Histogram, the nscores plots and the normal quantile plot for data generated from a right skewed distribution,week3,26,week3,27,Histogram, the nscores plots and the normal quantile plot for data generated from a left skewed distribution,week3,28,week3,29,Histogram, the ns

34、cores plots and the normal quantile plot for data generated from a uniform distribution (0,5),week3,30,week3,31,Looking at data - relationships,Two variables measured on the same individuals are associated if some values of one variable tend to occur more often with some values of the second variabl

35、e than with other values of that variable. When examining the relationship between two or more variables, we should first think about the following questions: What individuals do the data describe? What variables are present? How are they measured? Which variables are quantitative and which are cate

36、gorical? Is the purpose of the study is simply to explore the nature of the relationship, or do we hope to show that one variable can explain variation in the other?,week3,32,Response and explanatory variables,A response variable measure an outcome of a study. An explanatory variable explains or cau

37、ses changes in the response variables.Explanatory variables are often called independent variables and response variables are called dependent variables. The ides behind this is that response variables depend on explanatory variables.We usually call the explanatory variable x and the response variab

38、le y.,week3,33,Scatterplot,A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Each individual in the data appears as a point in the plot fixed by the values of both variables for that individual. Always plot the explanatory variable, if there is

39、 one, on the horizontal axis (the x axis) of a scatterplot. Examining and interpreting Scatterplots Look for overall pattern and striking deviations from that pattern. The overall pattern of a scatterplot can be described by the form, direction and strength of the relationship. An important kind of

40、deviation is an outlier, an individual value that falls outside the overall pattern.,week3,34,Example,There is some evidence that drinking moderate amounts of wine helps prevent heart attack. A data set contain information on yearly wine consumption (litters per person) and yearly deaths from heart

41、disease (deaths per 100,000 people) in 19 developed nations. Answer the following questions.What is the explanatory variable? What is the response variable? Examine the scatterplot below.,week3,35,week3,36,Interpretation of the scatterplot The pattern is fairly linear with a negative slope. No outli

42、ers. The direction of the association is negative . This means that higher levels of wine consumption are associated with lower death rates. This does not mean there is a causal effect. There could be lurking variables. For example, higher wine consumption could be linked to higher income, which wou

43、ld allow better medical care.MINITAB command for scatterplot Graph Plot,week3,37,Categorical variables in scatterplots,To add a categorical variable to a scatterplot, use a different colour or symbol for each category. The scatterplot below shows the relationship between the world record times for 1

44、0,000m run and the year for both men and women.,week3,38,Correlation,A sctterplot displays the form, direction and strength of the relationship between two quantitative variables. Correlation (denoted by r) measures the direction and strength of the liner relationship between two quantitative variab

45、les. Suppose that we have data on variables x and y for n individuals. The correlation r between x and y is given by,week3,39,Example,Family income and annual savings in thousand of $ for a sample of eight families are given below. savings income C3 C4 C51 36 -1.42887 -1.45101 2.073312 39 -1.02062 -

46、1.03144 1.052712 42 -0.61237 -0.61187 0.374695 45 -0.20412 -0.19230 0.039255 48 0.20412 0.22727 0.044336 51 0.61237 0.64684 0.396117 54 1.02062 1.06641 1.088408 56 1.42887 1.34612 1.92343Sum of C5 = 6.99429r = 6.99429/7 = 0.999185 MINITAB command: Stat Basic Statistics Correlation,week3,40,Propertie

47、s of correlation,Correlation requires both variables to be quantitative and make no use of the distinction between explanatory and response variables. Correlation r has no unit if measurement. Positive r indicates positive association between the variables and negative r indicates negative associati

48、on. Correlation measures the strength of only the linear relationship between two variables, it does not describe curved relationship! r is always a number between 1 and 1.Values of r near 0 indicates a weak linear relationship.The strength of the linear relationship increases as r moves away from 0

49、. Values of r close to 1 or 1 indicates that thepoints lie close to a straight line.r is not resistant. r is strongly affected by a few outliers.,week3,41,week3,42,Question from Term test, summer 99,MINITAB analyses of math and verbal SAT scores is given below.Variable N Mean Median TrMean StDev SE

50、MeanVerbal 200 595.65 586.00 595.57 73.21 5.18Math 200 649.53 649.00 650.37 66.35 4.69GPA 200 2.6300 2.6000 2.6439 0.5803 0.0410Variable Minimum Maximum Verbal 361.00 780.00 Math 441.00 800.00 GPA 0.3000 3.9000 Stem-and-leaf of Verbal N = 200Leaf Unit = 101 3 64 4 03419 4 56688888888999952 5 000000122222222333333333444444444(56) 5 5555555555555666666677777777777777888888888888888999999992 6 0000000001111111122222233333333344444444444444445 6 55555566666666677888888888999915 7 00111122445 7 55568,