1. Basic Statistics II Biostatistics, MHA, CDC, Jul 09
Prof. KG Satheesh Kumar
Asian School of Business

2. Frequency Distribution and Probability Distribution
Frequency Distribution: Plot of frequency along y-axis and variable along the x-axis
Histogram is an example
Probability Distribution: Plot of probability along y-axis and variable along x-axis
Both have same shape
Properties of probability distributions
Probability is always between 0 and 1
Sum of probabilities must be 1

3. Theoretical Probability Distributions
For a discrete variable we have discrete probability distribution
Binomial Distribution
Poisson Distribution
Geometric Distribution
Hypergeometric Distribution
For a continuous variable we have continuous probability distribution
Uniform (rectangular) Distribution
Exponential Distribution
Normal Distribution

4. The Normal Distribution
If a random variable, X is affected by many independent causes, none of which is overwhelmingly large, the probability distribution of X closely follows normal distribution. Then X is called normal variate and we write X ~ N(, 2), where  is the mean and 2 is the variance
A Normal pdf is completely defined by its mean,  and variance, 2. The square root of variance is called standard deviation .
If several independent random variables are normally distributed, their sum will also be normally distributed with mean equal to the sum of individual means and variance equal to the sum of individual variances.

5. The Normal pdf

6. The area under any pdf between two given values of X is the probability that X falls between these two values

7. Standard Normal Variate, Z
SNV, Z is the normal random variable with mean 0 and standard deviation 1
Tables are available for Standard Normal Probabilities
X and Z are connected by:
Z = (X - ) /  and X =  + Z
The area under the X curve between X1 and X2 is equal to the area under Z curve between Z1 and Z2.

8. Standard Normal Probabilities (Table of z distribution)
Standard Normal Probabilities (Table of z distribution)
The z-value is on the left and top margins and the probability (shaded area in the diagram) is in the body of the table

9. Illustration
Q.A tube light has mean life of 4500 hours with a standard deviation of 1500 hours. In a lot of 1000 tubes estimate the number of tubes lasting between 4000 and 6000 hours
P(4000<X<6000) = P(-1/3<Z<1) = =
Hence the probable number of tubes in a lot of 1000 lasting 4000 to 6000 hours is 472

10. Illustration
Q. Cost of a certain procedure is estimated to average Rs.25,000 per patient. Assuming normal distribution and standard deviation of Rs.5000, find a value such that 95% of the patients pay less than that.
Using tables, P(Z<Z1) = 0.95 gives Z1 = Hence X1 = x 5000 = Rs.33,225
95% of the patients pay less than Rs.33,225

11. Sampling Basics
Population or Universe is the collection of all units of interest. E.g.: Households of a specific type in a given city at a certain time. Population may be finite or infinite
Sampling Frame is the list of all the units in the population with identifications like Sl.Nos, house numbers, telephone nos etc
Sample is a set of units drawn from the population according to some specified procedure
Unit is an element or group of elements on which observations are made. E.g. a person, a family, a school, a book, a piece of furniture etc.

12. Census Vs Sampling Census Sampling
Thought to be accurate and reliable, but often not so if the population is large
More resources (money, time, manpower)
Unsuitable for destructive tests
Sampling
Less resources
Highly qualified and skilled persons can be used
Sampling error, which can be reduced using large and representative sample

13. Sampling Methods Probability Sampling (Random Sampling)
Simple Random Sampling
Systematic Random Sampling
Stratified Random Sampling
Cluster Sampling (Single stage , Multi-stage)
Non-probability Sampling
Convenience Sampling
Judgment Sampling
Quota Sampling

14. Limitations of Non-Random Sampling
Selection does not ensure a known chance that a unit will be selected (i.e. non-representative)
Inaccurate in view of the selection bias
Results cannot be used for generalisation because inferential statistics requires probability sampling for valid conclusions
Useful for pilot studies and exploratory research

15. Sampling Distribution and Standard Error of the Mean
The sampling distribution of x is the probability distribution of all possible values of x for a given sample size n taken from the population.
According to the Central Limit Theorem, for large enough sample size, n, the sampling distribution is approximately normal with mean  and standard deviation /n. This standard deviation is called standard error of the mean.
CLT holds for non-normal populations also and states: For large enough n, x ~ N(, 2/n)

16. Illustration
Q. When sampling from a population with SD 55, using a sample size of 150, what is the probability that the sample mean will be at least 8 units away from the population mean?
Standard Error of the mean, SE = 55/sqrt(150) =
Hence 8 units = SE
Area within SE on both sides of the mean = 2 * = 0.925
Hence required probability = = 0.075

17. Illustration
Q. An Economist wishes to estimate the average family income in a certain population. The population SD is known to be $4,500 and the economist uses a random sample of size 225. What is the probability that the sample mean will fall within $800 of the population mean?

18. Point and Interval Estimation
The value of an estimator (see next slide), obtained from a sample can be used to estimate the value of the population parameter. Such an estimate is called a point estimate.
This is a 50:50 estimate, in the sense, the actual parameter value is equally likely to be on either side of the point estimate.
A more useful estimate is the interval estimate, where an interval is specified along with a measure of confidence (90%, 95%, 99% etc)
The interval estimate with its associated measure of confidence is called a confidence interval.
A confidence interval is a range of numbers believed to include the unknown population parameter, with a certain level of confidence

19. Estimators
Population parameters (, 2, p) and Sample Statistics (x,s2, ps)
An estimator of a population parameter is a sample statistic used to estimate the parameter
Statistic,x is an estimator of parameter 
Statistic, s2 is an estimator of parameter 2
Statistic, ps is an estimator of parameter p

20. Illustration
Q. A wine importer needs to report the average percentage of alcohol in bottles of French wine. From experience with previous kinds of wine, the importer believes the population SD is 1.2%. The importer randomly samples 60 bottles of the new wine and obtains a sample mean of 9.3%. Find the 90% confidence interval for the average percentage of alcohol in the population.

21. Answer Standard Error = 1.2%/sqrt(60) = 0.1549%
For 90% confidence interval, Z = 1.645
Hence the margin of error = 1.645*0.1549%
= %
Hence 90% confidence interval is
9.3% +/- 0.3%

22. More Sampling Distributions
Sampling Distribution is the probability distribution of a given test statistic (e.g. Z), which is a numerical quantity calculated from sample statistic
Sampling distribution depends on the distribution of the population, the statistic being considered and the sample size
Distribution of Sample Mean: Z or t distribution
Distribution of Sample Proportion: Z (large sample)
Distribution of Sample Variance: Chi-square distribution

23. The t-distribution
The t-distribution is also bell-shaped and very similar to the Z(0,1) distribution
Its mean is 0 and variance is df/(df-2)
df = degrees of freedom = n-1 & n = sample size
For large sample size, t &Z are identical
For small n, the variance of t is larger than that of Z and hence wider tails, indicating the uncertainty introduced by unknown population SD or smaller sample size n

25. Illustration
Q. A large drugstore wants to estimate the average weekly sales for a brand of soap. A random sample of 13 weeks gives the following numbers: 123, 110, 95, 120, 87, 89, 100, 105, 98, 88, 75, 125, 101. Determine the 90% confidence interval for average weekly sales.
Sample mean = and Sample SD = From t-table, for 90% confidence at df = 12 is t = Hence Margin of Error = * 15.13/sqrt(13) = The 90% confidence interval is (93.75,108.71)

26. Chi-Square Distribution
Chi-square distribution is the probability distribution of the sum of several independent squared Z variables
It has a df parameter associated with it (like t distribution).
Being a sum of squares, the chi-squares cannot be negative and hence the distribution curve is entirely on the positive side, skewed to the right.
The mean is df and variance is 2df

27. Confidence Interval for population variance using chi-square distribution
A random sample of 30 gives a sample variance of 18,540 for a certain variable. Give a 95% confidence interval for the population variance
Point estimate for population variance = 18,540
Given df = 29, excel gives chi-square values:
For 2.5%, 45.7 and for 97.5, 16.0
Hence for the population variance,
the lower limit of the confidence interval
= *29/45.7 = 11,765 and
the upper limit of the confidence interval
= 18540*29/16.0 = 33,604

28. Chi-Square Distribution
Chi-square distribution is the probability distribution of the sum of several independent squared Z variables
It has a df parameter associated with it (like t distribution).
Being a sum of squares, the chi-squares cannot be negative and hence the distribution curve is entirely on the positive side, skewed to the right.
The mean is df and variance is 2df

29. Chi-Square Test for Goodness of Fit
A goodness-of-fit is a statistical test of how sample data support an assumption about the distribution of a population
Chi-square statistic used is
Χ2 = ∑(O-E)2/E, where O is the observed value and E the expected value
The above value is then compared with the critical value (obtained from table or using excel) for the given df and the required level of significance, α (1% or 5%)

30. Illustration
Q. A company comes out with a new watch and wants to find out whether people have special preferences for colour or whether all four colours under consideration are equally preferred. A random sample of 80 prospective buyers indicated preferences as follows: 12, 40, 8, 20. Is there a colour preference at 1% significance?
Assuming no preference, the expected values would all be 20. Hence the chi-square value is 64/ / / = 30.4
For df = 3 and 1% significance, the right tail area is 11.3.
The computed value of 30.4 is far greater than 11.3 and hence deeply in the rejection region. So we reject the assumption of no colour preference.

31. Q. Following data is about the births of new born babies on various days of the week during the past one year in a hospital. Can we assume that birth is independent of the day of the week? Sun:116, Mon:184, Tue: 148, Wed: 145, Thu: 153, Fri: 150, Sat: 154 (Total: 1050)
Ans: Assuming independence, the expected values would all be 1050/7 = 150. Hence the chi-square value is 342/ /150+22/150+52/150+32/150+42/150=2366/150 = 15.77
For df = 6 and 5% significance, the right tail area is 12.6.
The computed value of is greater than the critical value of 12.6 and hence falls in the rejection region. So we reject the assumption of independence.

32. Correlation
Correlation refers to the concomitant variation between two variables in such a way that change in one is associated with a change in the other
The statistical technique used to analyse the strength and direction of the above association between two variables is called correlation analysis

33. Correlation and Causation
Even if an association is established between two variables no cause-effect relationship is implied
Association between x and y may be looked upon as:
x causes y
y causes x
x and y influence each other (mutual influence)
x and y are both influenced by z, v (influence of third variable)
due to chance (spurious association)
Hence caution needed while interpreting correlation

34. Types of Correlations Positive (direct) and negative (inverse)
Positive: direction of change is the same
Negative: direction of change is opposite
Linear and non-linear
Linear: changes are in a constant ratio
Non-linear: ratio of change is varying
Simple, Partial and Multiple
Simple: Only two variables are involved
Partial: There may be third and other variables, but they are kept constant
Multiple: Association of multiple variables considered simultaneously

35. Scatter Diagrams Correlation coefficient r = 1 r = - 0.54 r = 0.85

36. Correlation Coefficient
Correlation coefficient (r) indicates the strength and direction of association
The value of r is between -1 and +1
-1: perfect negative correlation
+1: perfect positive correlation
Above 0.75: Very high correlation
0.50 to 0.75: High correlation
0.25 to 0.50: Low correlation
Below 0.25: Very low correlation

37. Methods of Correlation Analysis
Scatter Diagram
A quick approximate visual idea of association
Karl Pearson’s Coefficient of Correlation
For numeric data measured on interval or ratio scale
r = Cov(x,y) /(SDx SDy)
Spearman’s Rank Correlation
For ordinal (rank) data
R = 1 – 6 * Sum of Squared Difference of Ranks / [n(n2-1)]
Method of Least Squares
r2 = bxy * byx, i.e. product of regression coefficients

38. Karl Pearson Correlation Coefficient (Product-Moment Correlation)
r = Covariance (x,y) / (SD of x * SD of y)
Recall: n Var(X) = SSxx, nVar(Y) = SSYY and n Cov(X,Y) = SSXY
Thus r2 = Cov2(X,Y)/[Var(X) Var(Y)]
= SS2XY / (SSxx * SSYY)
Note: r2 is called coefficient of determination

39. Sample Problem
The following data refers to two variables, promotional expense (Rs. Lakhs) and sales (‘000 units) collected in the context of a promotional study. Calculate the correlation coefficient
Promo
Sales

40. Promo (X)
Sales (Y)
X - Ave(X)
Y - Ave (Y)
Sxy
Sxx
Syy 7 12 2 4 10 14 3 9 16 13 6 5 -3 -5 15 25 11 20 -2 -4 -6 24 36 83 58
124
Ave(X)
Ave(Y)
SSxy
SSxx
SSyy
Coefficient of Determination, r-squared = 83*83 / (58*124) =
Coefficient of Correlation, r = square root of =

41. Spearman’s Rank Correlation Coefficient
The ranks of 15 students in two subjects A and B are given below. Find Spearman’s Rank Correlation Coefficient
(1,10); (2,7); (3,2); (4,6); (5,4); (6,8); (7,3); (8,1); (9,11); (10,15); (11,9); (12,5); (13,14); (14,12) and (15,13)
Solution: SSD of Ranks = = 272
R = 1 – 6*272/(14*15*16) =
Hence moderate degree of positive correlation between the ranks of students in the two subjects

42. Regression Analysis
Statistical technique for expressing the relationship between two (or more) variables in the form of an equation (regression equation)
Dependent or response or predicted variable
Independent or regressor or predictor variable
Used for prediction or forecasting

43. Types of Regression Models
Simple and Multiple Regression Models
Simple: Only one independent variable
Multiple: More than one independent variable
Linear and Nonlinear Regression Models
Linear: Value of response variable changes in proportion to the change in predictor so that Y = a+bX

44. Simple Linear Regression Model
Y = a + bX,
a and b are constants to be determined using the given data
Note: More strictly, we may say: Y = ayx + byxX
To determine a and b solve the following two equations (called “normal equations”):
∑Y = a n + b ∑x (1)
∑YX = a ∑x + b ∑x (2)

45. Calculating Regression Coeff
Instead of solving the simultaneous equations one may directly use formulae
For Y = a + bX, i.e. regression of Y on X
byx = SSxy / SSxx
ayx = Y – byxX where mean values of Y, X are used
For X = a + bY form (regression of X on Y)
bxy = SSxy / SSyy
axy = Y – bxyX where mean values of Y, X are used

46. Example
For the earlier problem of Sales (dependent variable) Vs Promotional expenses (independent variable) set up the simple linear regression model and predict the sales when promotional spending is Rs.13 lakhs
Solution: We need to find a and b in Y = a + bX
b = SSxy / SSxx = 83/58 =
a = Y - bX, at mean = 10 – *7 =
Hence regression equation is Y = X
For X = 13 Lakhs, we get Y = 18.59, i.e. 18,590 units of predicted sales

47. Linear Regression using Excel

48. Properties of Regression Coeff
Coefficient of determination r2 = byx * bxy
If one regression coefficient is greater than one the other is less than one because r2 lies between 0 and 1
Both regression coeff must have the same sign, which is also the sign of the correlation coeff r
The regression lines intersect at the means of X and Y
Each regression coefficient gives the slope of the respective regression line

49. Coefficient of Determination
Recall:
SSyy = Sum of squared deviations of Y from the mean
Let us define
SSR as sum of squared deviations of estimated (using regression equation) values of Y from the mean
SSE as the sum of squared deviations of errors (error means actual Y ~ estimated Y)
It can be shown that:
SSyy = SSR + SSE, i.e. Total Variation = Explained Variation + Unexplained (error) Variation
r2 = SSR/SSyy = Explained Variation / Total Variation
Thus r2 represents the proportion of the total variability of the dependent variable y that is accounted for or explained by the independent variable x

50. Coefficient of Determination for Statistical Validity of Promo-Sales Regression Model
Promo (X)
Sales (Y)
Ye = X
Squared deviation of Ye from Mean
Squared deviation of Ye from Y
Squared Deviation of Y from Mean 7 12
10.00
0.00
4.00 4 10 14
14.29
18.43
0.09 16 9 13
12.86
8.19
0.02 5
5.71
0.50 25 11 15
15.72
32.76
0.52
7.14 3
4.28
0.08 36
118.77
5.22
124
SSR
SSE
Ssyy
Coefficient of determination, r-squared = /124 =
Thus 96% of the variation in sales is explained by promo expenses