Here are the key steps to solve this problem: 1) The population mean lifetime for manufacturer A's bulbs is μ = 140 hours 2) We are sampling n = 20 bulbs 3) The sampling distribution of the sample mean (x-bar) is Normal with: - Mean = Population mean (μ) = 140 hours - Standard deviation = Population standard deviation (σ) / √n = Let's assume σ = 10 hours Then standard deviation of x-bar = 10/√20 = 2.22 hours 4) We want P(x-bar < 138) 5) Standardize: (138 - 140) / 2.22 = -0.89 6)

1. Introduction to Probability and Statistics
9th Week (5/10)
1. Descriptive Statistics
2. Sampling Theory

2. Probability is a science of ( ).
Statistics is a science of ( ).

3. Probability is the Science of Uncertainty.
 It is used by Physicists to predict the behaviour of
elementary particles.
 It is used by engineers to build computers.
 It is used by economists to predict the behaviour of the
economy.
 It is used by stockbrokers to make money on the
stockmarket.
 It is used by psychologists to determine if you should get
that job.

4. What about Statistics?
 Statistics is the Science of Data.
 There are two kinds of statistics
 Descriptive Statistics: Discipline of quantitatively
describing the main features of a collection of data
 Inferential Statistics: It is a discipline that allows us
to estimate unknown quantities by making some
elementary measurements. Using these estimates we
can then make Predictions and Forecast the Future

5. Descriptive Statistics
• Describing data with tables and graphs
(quantitative or categorical variables)
• Numerical descriptions of center, variability,
position (quantitative variables)
• Bivariate descriptions (In practice, most studies
have several variables)

6. 1. Tables and Graphs
Frequency distribution: Lists possible values of
variable and number of times each occurs
Example: Student survey (n = 60)
“political ideology” measured as ordinal variable
with 1 = very liberal, …, 4 = moderate, …, 7 =
very conservative

8. Histogram: Bar graph of
frequencies or percentages

9. Shapes of histograms
(for quantitative variables)
• Bell-shaped (IQ, SAT, political ideology in all U.S. )
• Skewed right (annual income, no. times arrested)
• Skewed left (score on easy exam)
• Bimodal (polarized opinions)

10. Stem-and-leaf plot (John Tukey, 1977)
Example: Exam scores (n = 40 students)
Stem Leaf
3 6
4
5 37
6 235899
7 011346778999
8 00111233568889
9 02238

11. 2.Numerical descriptions
Let y denote a quantitative variable, with
observations y1 , y2 , y3 , … , yn
a. Describing the center
Median: Middle measurement of ordered sample
Mean:
y1 + y2 + ... + yn Σyi
y= =
n n

12. Properties of mean and median
• For symmetric distributions, mean = median
• For skewed distributions, mean is drawn in
direction of longer tail, relative to median
• Mean valid for interval scales, median for
interval or ordinal scales
• Mean sensitive to “outliers” (median often
preferred for highly skewed distributions)
• When distribution symmetric or mildly skewed or
discrete with few values, mean preferred
because uses numerical values of observations

13. Examples:
• New York Yankees baseball team, 2006
mean salary = $7.0 million
median salary = $2.9 million
How possible? Direction of skew?
• Give an example for which you would expect
mean < median

14. b. Describing variability
Range: Difference between largest and smallest
observations
(but highly sensitive to outliers, insensitive to shape)
Standard deviation: A “typical” distance from the mean
The deviation of observation i from the mean is
yi − y

15. The variance of the n observations is
Σ ( yi − y ) ( y1 − y ) + ... + ( yn − y )
2 2 2
s =
2
=
n−1 n−1
The standard deviation s is the square root of the variance,
s = s 2

16. Example: Political ideology
• For those in the student sample who attend religious
services at least once a week (n = 9 of the 60),
• y = 2, 3, 7, 5, 6, 7, 5, 6, 4
y = 5.0,
(2 − 5) 2 + (3 − 5) 2 + ... + (4 − 5) 2 24
s2 = = = 3.0
9 −1 8
s = 3.0 = 1.7
For entire sample (n = 60), mean = 3.0, standard deviation = 1.6,
tends to have similar variability but be more liberal

17. c. Measures of position
pth percentile: p percent of observations
below it, (100 - p)% above it.
 p = 50: median
 p = 25: lower quartile (LQ)
 p = 75: upper quartile (UQ)
 Interquartile range IQR = UQ - LQ

18. Quartiles portrayed graphically by box plots
(John Tukey)
Example: weekly TV watching for n=60 from
student survey data file, 3 outliers

19. Box plots have box from LQ to UQ, with
median marked. They portray a five-
number summary of the data:
Minimum, LQ, Median, UQ, Maximum
except for outliers identified separately
Outlier = observation falling
below LQ – 1.5(IQR)
or above UQ + 1.5(IQR)
Ex. If LQ = 2, UQ = 10, then IQR = 8 and
outliers above 10 + 1.5(8) = 22

20. 3. Bivariate description
• Usually we want to study associations between two or
more variables (e.g., how does number of close
friends depend on gender, income, education, age,
working status, rural/urban, religiosity…)
• Response variable: the outcome variable
• Explanatory variable(s): defines groups to compare
Ex.: number of close friends is a response variable,
while gender, income, … are explanatory variables
Response var. also called “dependent variable”
Explanatory var. also called “independent variable”

21. Summarizing associations:
• Categorical var’s: show data using contingency tables
• Quantitative var’s: show data using scatterplots
• Mixture of categorical var. and quantitative var. (e.g.,
number of close friends and gender) can give
numerical summaries (mean, standard deviation) or
side-by-side box plots for the groups
• Ex. General Social Survey (GSS) data
Men: mean = 7.0, s = 8.4
Women: mean = 5.9, s = 6.0
Shape? Inference questions for later chapters?

22. Example: Income by highest degree

23. Contingency Tables
• Cross classifications of categorical variables in
which rows (typically) represent categories of
explanatory variable and columns represent
categories of response variable.
• Counts in “cells” of the table give the numbers of
individuals at the corresponding combination of
levels of the two variables

24. Happiness and Family Income
(GSS 2008 data: “happy,” “finrela”)
Happiness
Income Very Pretty Not too Total
-------------------------------
Above Aver. 164 233 26 423
Average 293 473 117 883
Below Aver. 132 383 172 687
------------------------------
Total 589 1089 315 1993

25. Can summarize by percentages on response
variable (happiness)
Example: Percentage “very happy” is
39% for above aver. income (164/423 = 0.39)
33% for average income (293/883 = 0.33)
19% for below average income (??)

26. Happiness
Income Very Pretty Not too Total
--------------------------------------------
Above 164 (39%) 233 (55%) 26 (6%) 423
Average 293 (33%) 473 (54%) 117 (13%) 883
Below 132 (19%) 383 (56%) 172 (25%) 687
----------------------------------------------
Inference questions for later chapters? (i.e., what can
we conclude about the corresponding population?)

27. Scatterplots (for quantitative variables)
plot response variable on vertical axis,
explanatory variable on horizontal axis
Example: Table 9.13 (p. 294) shows UN data for several
nations on many variables, including fertility (births per
woman), contraceptive use, literacy, female economic
activity, per capita gross domestic product (GDP), cell-
phone use, CO2 emissions
Data available at
http://www.stat.ufl.edu/~aa/social/data.html

29. Example: Survey in Alachua County, Florida,
on predictors of mental health
(data for n = 40 on p. 327 of text and at
www.stat.ufl.edu/~aa/social/data.html)
y = measure of mental impairment (incorporates various
dimensions of psychiatric symptoms, including aspects of
depression and anxiety)
(min = 17, max = 41, mean = 27, s = 5)
x = life events score (events range from severe personal
disruptions such as death in family, extramarital affair, to
less severe events such as new job, birth of child, moving)
(min = 3, max = 97, mean = 44, s = 23)

31. Bivariate data from 2000 Presidential election
Butterfly ballot, Palm Beach County, FL, text p.290

32. Example: The Massachusetts Lottery
(data for 37 communities)
% income
spent on
lottery
Per capita income

33. Correlation describes strength of
association
• Falls between -1 and +1, with sign indicating direction
of association (formula later in Chapter 9)
The larger the correlation in absolute value, the stronger
the association (in terms of a straight line trend)
Examples: (positive or negative, how strong?)
Mental impairment and life events, correlation =
GDP and fertility, correlation =
GDP and percent using Internet, correlation =

35. Inferential Statistics: Fortune Teller
 How can she read the future?
 Analysis of Data from Her Previous Victims (Clients)
 Make Hypotheses
 Test Them
 Fool You!

36. Population and Sample
- Often in practice we are interested in drawing valid conclusions
about a large group of individuals or objects.
- Instead of examining the entire group, called the population, which
may be difficult or impossible to do, we may examine only a small
part of this population, which is called a sample.
- The process of obtaining samples is called sampling.
Sampling
population
Sample

37. Statistical Inference
- We do this with the aim of inferring certain facts about the
population from results found in the sample, a process known as
statistical inference.

38. Sampling With and Without Replacement
- Population may be finite or infinite.
- If finite, Sampling method is important.
- If we draw an object from an urn, we have the choice of replacing
or not replacing the object into the urn before we draw again.
- Sampling with replacement: Sampling where each member of a
population may be chosen more than once
- Sampling without replacement: sampling where each member
cannot be chosen more than once
- A finite population that is sampled with replacement can
theoretically be considered infinite since samples of any size can
be drawn without exhausting the population.

39. Random Samples, Random Numbers
- Clearly, the reliability of conclusions drawn concerning a population
depends on whether the sample is properly chosen so as to
represent the population sufficiently well, and one of the important
problems of statistical inference is just how to choose a sample.
- One way to do this for finite populations is to make sure that each
member of the population has the same chance of being in the
sample, which is then often called a random sample.
- Random sampling can be accomplished for relatively small
populations by drawing lots or, equivalently, by using a table of
random numbers specially constructed for such purposes.
- Because inference from sample to population cannot be
certain, we must use the language of probability in any
statement of conclusions.

40. Random Samples, Random Numbers

41. Population Parameters
- A population is considered to be known when we know the
probability distribution f (x) (probability function or density function)
of the associated random variable X.
- If X is a random variable whose values are the heights (or weights)
of the 12,000 students, then X has a probability distribution f (x).
- If, for example, X is normally distributed, we say that the population
is normally distributed or that we have a normal population.
Similarly, if X is binomially distributed, we say that the population is
binomially distributed or that we have a binomial population.
- There will be certain quantities that appear in f(x), such as µ and σ
in the case of the normal distribution or p in the case of the
binomial distribution.
- Other quantities such as the median, moments, and skewness can
then be determined in terms of these.

42. Population Parameters
- All such quantities are often called population parameters.
- When we are given the population so that we know f(x), then the
population parameters are also known.
- An important problem arises when the probability distribution
f(x) of the population is not known precisely, although we may
have some idea of, or at least be able to make some hypothesis
concerning, the general behavior of f(x).
- For example, we may have some reason to suppose that a
particular population is normally distributed.
- In that case we may not know one or both of the values and so we
might wish to draw statistical inferences about them.

43. Population Parameters

44. Sample Statistics
- We can take random samples from the population and then use
these samples to obtain values that serve to estimate and test
hypotheses about the population parameters.

45. Sample Statistics

46. Sample statistics /
Population parameters
• We distinguish between summaries of samples
(statistics) and summaries of populations
(parameters).
• Common to denote statistics by Roman letters,
parameters by Greek letters:

47. Sample Statistics
- In general, corresponding to each population parameter there will
be a statistic to be computed from the sample.
- Usually the method for obtaining this statistic from the sample is
similar to that for obtaining the parameter from a finite population,
since a sample consists of a finite set of values.
- As we shall see, however, this may not always produce the “best
estimate,” and one of the important problems of sampling theory is
to decide how to form the proper sample statistic that will best
estimate a given population parameter.
- Where possible we shall try to use Greek letters, such µ as σ and ,
for values of population parameters, and Roman letters, m, s, etc.,
for values of corresponding sample statistics.

48. Sampling Distribution
- As we have seen, a sample statistic that is computed from X1, . . . ,
Xn is a function of these random variables and is therefore itself a
random variable.
- The probability distribution of a sample statistic is often called the
sampling distribution of the statistic.
- Alternatively we can consider all possible samples of size n that
can be drawn from the population, and for each sample we
compute the statistic.
- In this manner we obtain the distribution of the statistic, which is its
sampling distribution.
- For a sampling distribution, we can of course compute a mean,
variance, standard deviation, moments, etc.
- The standard deviation is sometimes also called the standard error.

49. The Sample Mean

50. Sampling Distribution of Means

51. Sampling Distribution of Means

52. Sampling Distribution of Means

53. Sampling Distribution of Proportions
- Suppose that a population is infinite and binomially distributed, with
p and q = 1- p being the respective probabilities that any given
member exhibits or does not exhibit a certain property.
- Consider all possible samples of size n drawn from this population,
and for each sample determine the statistic that is the proportion P
of successes.
- In the case of the coin, p would be the proportion of heads turning
up in n tosses. Then we obtain a sampling distribution of
proportions whose mean µp and standard deviation σp are given by
- For finite populations in which sampling is without replacement, the
second equation in (9) is replaced by as given by (6) with

54. Sampling Distribution of Proportions

55. Example 6. Find the probability that in 120 tosses of a fair coin (a) between 40%
and 60% will be heads, (b) or more will be heads.

56. Example 6. (Continue)

57. Sampling Distribution of Differences
and Sums

58. Sampling Distribution of Differences
and Sums

59. Example 7. The electric light bulbs of manufacturer A have a mean lifetime of
1400 hours with a standard deviation of 200 hours, while those of manufacturer
B have a mean lifetime of 1200 hours with a standard deviation of 100 hours. If
random samples of 125 bulbs of each brand are tested, what is the probability
that the brand A bulbs will have a mean lifetime that is at least (a) 160 hours, (b)
250 hours more than the brand B bulbs?

60. Example 7. (Answer)

61. Example 8. Ball bearings of a given brand weigh 0.50 oz with a standard deviation of
0.02 oz. What is the probability that two lots, of 1000 ball bearings each, will differ in
weight by more than 2 oz?

62. The Sample Variance

63. The Sample Variance

64. Sampling Distribution of Variances

65. (Continuous) Chi Square Distribution
A special gamma distribution α = r/2, β = 2
 PD
 E(X)
 Var(X)

66. (Continuous) Chi Square Distribution

67. (Continuous) Chi Square Distribution

68. Case Where Population Variance Is Unknown

69. (Continuous) Student t Distribution

70. (Continuous) Student t Distribution

71. (Continuous) F Distribution

72. (Continuous) F Distribution

73. (Continuous) F Distribution
⊙ Effect of the degree of freedom
For α, 100(1- α)% : fα(m, n)
(1) P(X ≥ fα(m, n) ) = α
(2) P(f1-α/2(m, n) ≤ X ≤ fα/2(m, n)) = α
(3) F ∼ F (m, n) ⇒ 1/F ∼ F (n, m)

74. (Continuous) F Distribution

75. Sampling Distribution of Ratios of Variances

76. Other Statistics

77. Other Statistics